Single Best Answer MCQs for Sub-Topics of Topic 12: Vectors & Three-Dimensional Geometry

Question 1. Which of the following is a scalar quantity?

(A) Displacement

(B) Velocity

(C) Mass

(D) Acceleration

Question 2. A vector has magnitude 5 units and makes an angle of $120^\circ$ with the positive x-axis. What is its representation in terms of components?

(A) $5(\cos 120^\circ \hat{i} + \sin 120^\circ \hat{j})$

(B) $5(\cos 120^\circ \hat{j} + \sin 120^\circ \hat{i})$

(C) $5(\cos 30^\circ \hat{i} + \sin 30^\circ \hat{j})$

(D) $5(\cos 60^\circ \hat{i} + \sin 60^\circ \hat{j})$

Question 3. The magnitude of the vector $\vec{v} = 3\hat{i} - 4\hat{j}$ is:

(A) 1

(B) 5

(C) 7

(D) 25

Question 4. The direction of a vector is represented by:

(A) Its magnitude

(B) A single number

(C) An angle or direction cosines

(D) Its initial point

Question 5. A vector whose initial and terminal points coincide is called a:

(A) Unit vector

(B) Zero vector

(C) Coinitial vector

(D) Collinear vector

Question 6. Which of the following statements is true about collinear vectors?

(A) They must have the same magnitude.

(B) They must have the same direction.

(C) They lie on the same line or parallel lines.

(D) They must have the same initial point.

Question 7. Two vectors $\vec{a}$ and $\vec{b}$ are equal if:

(A) They have the same magnitude.

(B) They have the same direction.

(C) They have the same initial point.

(D) They have the same magnitude and direction.

Question 8. If $\vec{a} = 2\hat{i} + 3\hat{j}$ and $\vec{b} = -\hat{i} + 5\hat{j}$, find $\vec{a} + \vec{b}$.

(A) $3\hat{i} + 8\hat{j}$

(B) $\hat{i} + 8\hat{j}$

(C) $\hat{i} - 2\hat{j}$

(D) $3\hat{i} - 2\hat{j}$

Question 9. If $\vec{u} = 4\hat{i} - 2\hat{j}$ and $\vec{v} = \hat{i} + \hat{j}$, find $2\vec{u} - 3\vec{v}$.

(A) $5\hat{i} - 7\hat{j}$

(B) $7\hat{i} - 7\hat{j}$

(C) $5\hat{i} - 5\hat{j}$

(D) $7\hat{i} - 5\hat{j}$

Question 10. If $\vec{a}$ is a non-zero vector and $k$ is a scalar, then $k\vec{a}$ is a unit vector if:

(A) $k=1$

(B) $|k| = |\vec{a}|$

(C) $|k| |\vec{a}| = 1$

(D) $k = \frac{1}{|\vec{a}|}$

Question 11. The associative property of vector addition states that for any vectors $\vec{a}, \vec{b}, \vec{c}$:

(A) $\vec{a} + \vec{b} = \vec{b} + \vec{a}$

(B) $(\vec{a} + \vec{b}) + \vec{c} = \vec{a} + (\vec{b} + \vec{c})$

(C) $\vec{a} + \vec{0} = \vec{a}$

(D) $k(\vec{a} + \vec{b}) = k\vec{a} + k\vec{b}$

Question 12. For any vector $\vec{v}$, $\vec{v} + (-\vec{v})$ equals:

(A) $\vec{v}$

(B) $2\vec{v}$

(C) $\vec{0}$

(D) $-\vec{v}$

Question 13. If $k_1$ and $k_2$ are scalars, and $\vec{a}$ is a vector, then $(k_1 + k_2)\vec{a} =$:

(A) $k_1 \vec{a} k_2 \vec{a}$

(B) $k_1 \vec{a} + k_2 \vec{a}$

(C) $(k_1 k_2)\vec{a}$

(D) $\vec{a}(k_1 + k_2)$

Question 14. The magnitude of a scalar multiple $k\vec{a}$ is given by:

(A) $k |\vec{a}|$

(B) $|k| |\vec{a}|$

(C) $k^2 |\vec{a}|$

(D) $|k|^2 |\vec{a}|$

Question 15. The vector $\vec{AB}$ refers to a vector starting from point A and ending at point B. What is the relationship between $\vec{AB}$ and $\vec{BA}$?

(A) $\vec{AB} = \vec{BA}$

(B) $\vec{AB} = -\vec{BA}$

(C) $\vec{AB} = 2\vec{BA}$

(D) $|\vec{AB}| = -|\vec{BA}|$

Question 16. If vectors $\vec{a}$ and $\vec{b}$ are represented by adjacent sides of a parallelogram, their sum $\vec{a} + \vec{b}$ is represented by:

(A) The other adjacent side.

(B) The diagonal from the same initial point.

(C) The diagonal from the terminal point of $\vec{b}$ to the terminal point of $\vec{a}$.

(D) The line segment joining the midpoints of the sides.

Question 17. A vector that has magnitude 1 is called a:

(A) Zero vector

(B) Collinear vector

(C) Unit vector

(D) Equal vector

Question 18. If $\vec{a}$ is a vector, the vector $\frac{\vec{a}}{|\vec{a}|}$ is:

(A) A vector in the same direction as $\vec{a}$ with magnitude $|\vec{a}|$.

(B) A unit vector in the same direction as $\vec{a}$.

(C) A vector in the opposite direction as $\vec{a}$ with magnitude 1.

(D) The zero vector.

Question 19. For any scalar $k$, the vector $k\vec{0}$ (where $\vec{0}$ is the zero vector) is equal to:

(A) $k$

(B) $\vec{0}$

(C) $k\vec{0}$ (itself)

(D) Cannot be determined

Question 20. The process of multiplying a vector by a real number (scalar) is called:

(A) Vector addition

(B) Vector subtraction

(C) Scalar multiplication

(D) Dot product

Question 21. If $\vec{a}$ and $\vec{b}$ are two vectors, which geometric law is used for $\vec{a} + \vec{b}$ if they represent two sides of a triangle taken in order?

(A) Parallelogram Law

(B) Triangle Law

(C) Associative Law

(D) Commutative Law

Question 22. The zero vector has:

(A) Zero magnitude and a specific direction.

(B) Non-zero magnitude and a specific direction.

(C) Zero magnitude and an arbitrary direction.

(D) Non-zero magnitude and an arbitrary direction.

Question 23. If $\vec{a} = \vec{b}$, which of the following must be true?

(A) $|\vec{a}| = |\vec{b}|$ but directions can be different.

(B) Directions are the same but magnitudes can be different.

(C) Initial points are the same.

(D) $|\vec{a}| = |\vec{b}|$ and directions are the same.

Question 24. If $k$ is a negative scalar and $\vec{a}$ is a non-zero vector, then the direction of $k\vec{a}$ is:

(A) Same as $\vec{a}$.

(B) Opposite to $\vec{a}$.

(C) Perpendicular to $\vec{a}$.

(D) Independent of the direction of $\vec{a}$.

Question 25. For three vectors $\vec{a}, \vec{b}, \vec{c}$, the property $\vec{a} + (\vec{b} + \vec{c}) = (\vec{a} + \vec{b}) + \vec{c}$ is known as:

(A) Commutativity

(B) Associativity

(C) Distributivity

(D) Identity property

Question 26. If $\vec{a} = 5\hat{i}$ and $\vec{b} = -5\hat{i}$, then $\vec{a}$ and $\vec{b}$ are:

(A) Equal vectors

(B) Unit vectors

(C) Coinitial vectors

(D) Collinear vectors

Question 27. What is the result of $\vec{a} - \vec{a}$?

(A) $|\vec{a}|$

(B) $2\vec{a}$

(C) $\vec{0}$

(D) A scalar quantity

Question 28. The identity element for vector addition is:

(A) Any unit vector

(B) The zero vector

(C) The scalar 0

(D) The vector itself

Question 29. The inverse element for vector addition of vector $\vec{v}$ is:

(A) $\vec{v}$

(B) $-\vec{v}$

(C) $\frac{1}{|\vec{v}|}\vec{v}$

(D) $\vec{0}$

Question 30. The direction of the vector $-2\hat{i}$ is:

(A) Along the positive x-axis.

(B) Along the negative x-axis.

(C) Along the positive y-axis.

(D) Along the negative y-axis.

Question 1. The position vector of a point P with coordinates $(x, y, z)$ relative to the origin O is:

(A) $x+y+z$

(B) $x\hat{i} + y\hat{j} + z\hat{k}$

(C) $(x, y, z)$

(D) $\sqrt{x^2+y^2+z^2}$

Question 2. If point A has coordinates $(1, 2, 3)$ and point B has coordinates $(4, 5, 6)$, the vector $\vec{AB}$ is:

(A) $3\hat{i} + 3\hat{j} + 3\hat{k}$

(B) $-3\hat{i} - 3\hat{j} - 3\hat{k}$

(C) $5\hat{i} + 7\hat{j} + 9\hat{k}$

(D) $5\hat{i} + 7\hat{j} + 9\hat{k}$

Question 3. The magnitude of the vector $\vec{v} = 2\hat{i} - \hat{j} + 2\hat{k}$ in 3D space is:

(A) 3

(B) $\sqrt{7}$

(C) $\sqrt{5}$

(D) 5

Question 4. If $\vec{a} = 2\hat{i} - \hat{j}$ and $\vec{b} = 4\hat{i} - 2\hat{j}$, then $\vec{a}$ and $\vec{b}$ are:

(A) Perpendicular vectors

(B) Parallel vectors

(C) Unit vectors

(D) Equal vectors

Question 5. A vector $\vec{v}$ in 2D is given by components $(v_x, v_y)$. Its magnitude is given by:

(A) $v_x + v_y$

(B) $\sqrt{v_x + v_y}$

(C) $v_x^2 + v_y^2$

(D) $\sqrt{v_x^2 + v_y^2}$

Question 6. The unit vector in the direction of $\vec{a} = 3\hat{i} - 4\hat{j}$ is:

(A) $3\hat{i} - 4\hat{j}$

(B) $\frac{1}{5}(3\hat{i} - 4\hat{j})$

(C) $\frac{1}{7}(3\hat{i} - 4\hat{j})$

(D) $5(3\hat{i} - 4\hat{j})$

Question 7. If $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$, then the components of $\vec{r}$ are:

(A) $(|\vec{r}|, \text{direction})$

(B) $(x, y, z)$

(C) $(\hat{i}, \hat{j}, \hat{k})$

(D) $(x^2, y^2, z^2)$

Question 8. A vector $\vec{v}$ is a linear combination of vectors $\vec{v}_1, \vec{v}_2, \dots, \vec{v}_n$ if it can be written in the form:

(A) $\vec{v} = \vec{v}_1 + \vec{v}_2 + \dots + \vec{v}_n$

(B) $\vec{v} = c_1\vec{v}_1 + c_2\vec{v}_2 + \dots + c_n\vec{v}_n$, where $c_i$ are scalars.

(C) $\vec{v} = |\vec{v}_1|\vec{v}_1 + |\vec{v}_2|\vec{v}_2 + \dots + |\vec{v}_n|\vec{v}_n$

(D) $\vec{v} = \vec{v}_1 - \vec{v}_2 - \dots - \vec{v}_n$

Question 9. Three vectors $\vec{a}, \vec{b}, \vec{c}$ are coplanar if one of them can be expressed as a linear combination of the other two (provided the other two are non-collinear). If $\vec{c} = x\vec{a} + y\vec{b}$, where $x, y$ are scalars, does this guarantee coplanarity?

(A) Yes, always.

(B) Yes, provided $\vec{a}$ and $\vec{b}$ are non-collinear.

(C) No, only if $x=y=0$.

(D) No, this only implies collinearity.

Question 10. If $\vec{a} = \hat{i} - 2\hat{j}$ and $\vec{b} = -2\hat{i} + 4\hat{j}$, are $\vec{a}$ and $\vec{b}$ collinear?

(A) Yes, because $\vec{b} = -2\vec{a}$.

(B) Yes, because their magnitudes are related.

(C) No, because their components are different.

(D) Cannot be determined.

Question 11. In a 2D plane, any vector can be uniquely expressed as a linear combination of two non-zero, non-collinear vectors. These two vectors are called a:

(A) Unit basis

(B) Orthonormal basis

(C) Basis

(D) Collinear set

Question 12. The position vector of point $(0, 0, 0)$ is:

(A) 0

(B) 1

(C) $\hat{i}+\hat{j}+\hat{k}$

(D) $\vec{0}$

Question 13. If $\vec{a} = x\hat{i} + y\hat{j}$ is a unit vector, then:

(A) $x^2+y^2=0$

(B) $x^2+y^2=1$

(C) $x+y=1$

(D) $x=1, y=0$

Question 14. For what value of $k$ are the vectors $\vec{a} = 2\hat{i} - 3\hat{j}$ and $\vec{b} = k\hat{i} + 6\hat{j}$ collinear?

(A) 4

(B) -4

(C) $\frac{1}{4}$

(D) $-\frac{1}{4}$

Question 15. The components of a vector $\vec{PQ}$, where P is $(x_1, y_1, z_1)$ and Q is $(x_2, y_2, z_2)$, are:

(A) $(x_1-x_2, y_1-y_2, z_1-z_2)$

(B) $(x_1+x_2, y_1+y_2, z_1+z_2)$

(C) $(x_2-x_1, y_2-y_1, z_2-z_1)$

(D) $(\sqrt{(x_2-x_1)^2}, \sqrt{(y_2-y_1)^2}, \sqrt{(z_2-z_1)^2})$

Question 16. If $\vec{a} = 2\hat{i} - \hat{j} + \hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} - \hat{k}$, find the magnitude of $\vec{a} + \vec{b}$.

(A) $\sqrt{10}$

(B) $\sqrt{11}$

(C) $\sqrt{12}$

(D) $\sqrt{13}$

Question 17. Which of the following vectors is a unit vector?

(A) $\hat{i} + \hat{j}$

(B) $\frac{1}{\sqrt{2}}(\hat{i} - \hat{j})$

(C) $\hat{i} + \hat{j} + \hat{k}$

(D) $2\hat{i}$

Question 18. Can the vector $\vec{v} = 5\hat{i} - \hat{j}$ be expressed as a linear combination of $\vec{a} = \hat{i} + \hat{j}$ and $\vec{b} = \hat{i} - \hat{j}$?

(A) Yes, always.

(B) Yes, provided $\vec{a}$ and $\vec{b}$ are non-collinear.

(C) No, only in 3D.

(D) Only if $\vec{v}$ is a scalar multiple of $\vec{a}$ or $\vec{b}$.

Question 19. What are the components of a vector lying on the x-axis with magnitude 7?

(A) $(0, 7, 0)$

(B) $(7, 0, 0)$

(C) $(0, 0, 7)$

(D) $(7, 7, 7)$

Question 20. If $\vec{a} = \hat{i} + 2\hat{j}$ and $\vec{b} = x\hat{i} + y\hat{j}$ are such that $\vec{b} = 3\vec{a}$, what are the values of $x$ and $y$?

(A) $x=3, y=6$

(B) $x=1, y=2$

(C) $x=2, y=1$

(D) $x=6, y=3$

Question 21. The position vector of the origin is:

(A) $\hat{i}+\hat{j}+\hat{k}$

(B) $(0,0,0)$

(C) $\vec{0}$

(D) 0

Question 22. For points A(1, -1, 2) and B(2, 1, -1), the magnitude of vector $\vec{AB}$ is:

(A) $\sqrt{11}$

(B) $\sqrt{14}$

(C) $\sqrt{17}$

(D) $\sqrt{19}$

Question 23. Three points A, B, C are collinear if vector $\vec{AB}$ is a scalar multiple of vector $\vec{AC}$. If A=(1,2), B=(3,6), C=(4,8), are A, B, C collinear?

(A) Yes, because $\vec{AB} = (2,4)$ and $\vec{AC} = (3,6)$, and $(3,6) = 1.5(2,4)$.

(B) Yes, because $|\vec{AB}| + |\vec{BC}| = |\vec{AC}|$.

(C) No, they are not collinear.

(D) Yes, because their sum is zero.

Question 24. If $\vec{v} = c_1 \hat{i} + c_2 \hat{j} + c_3 \hat{k}$, the coefficients $c_1, c_2, c_3$ are called the ________ of the vector $\vec{v}$.

(A) Direction ratios

(B) Direction cosines

(C) Components

(D) Magnitudes

Question 25. Any vector $\vec{r}$ in 3D space can be uniquely written as $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$, where $\hat{i}, \hat{j}, \hat{k}$ form an orthogonal basis. The property of uniqueness depends on:

(A) The magnitude of $\vec{r}$.

(B) The basis vectors being unit vectors.

(C) The basis vectors being mutually orthogonal and non-zero.

(D) The origin of the coordinate system.

Question 26. The vector $\vec{a} = 2\hat{i} + 3\hat{j} - \hat{k}$ and $\vec{b} = -4\hat{i} - 6\hat{j} + 2\hat{k}$ are:

(A) Perpendicular

(B) Parallel

(C) Unit vectors

(D) Coplanar but not collinear

Question 27. If a vector makes angles $\alpha, \beta, \gamma$ with the positive x, y, z axes respectively, then $\cos \alpha, \cos \beta, \cos \gamma$ are called its:

(A) Direction ratios

(B) Direction angles

(C) Direction cosines

(D) Components

Question 28. Any two non-zero non-collinear vectors $\vec{a}$ and $\vec{b}$ in a plane form a basis for that plane. This means:

(A) Any vector in that plane can be written as $c\vec{a}$ for some scalar $c$.

(B) Any vector in that plane can be written as $c\vec{b}$ for some scalar $c$.

(C) Any vector in that plane can be uniquely written as $c_1\vec{a} + c_2\vec{b}$ for some scalars $c_1, c_2$.

(D) Only vectors parallel to $\vec{a}$ or $\vec{b}$ can be represented.

Question 29. If the position vectors of points A and B are $\vec{a}$ and $\vec{b}$ respectively, the position vector of point C on AB produced such that AC = 3AB is:

(A) $3\vec{b} - 2\vec{a}$

(B) $3\vec{a} - 2\vec{b}$

(C) $2\vec{b} - \vec{a}$

(D) $2\vec{a} - \vec{b}$

Question 30. The coordinates of the point with position vector $2\hat{i} - 5\hat{j} + \hat{k}$ are:

(A) $(2, 5, 1)$

(B) $(2, -5, 1)$

(C) $(-2, 5, -1)$

(D) $(5, -2, 1)$

Question 1. The scalar product of two vectors $\vec{a}$ and $\vec{b}$ is defined as $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta$, where $\theta$ is the angle between them. The dot product is:

(A) A vector quantity

(B) A scalar quantity

(C) Always positive

(D) Always zero

Question 2. If $\vec{a} = 2\hat{i} + 3\hat{j}$ and $\vec{b} = \hat{i} - \hat{j}$, find $\vec{a} \cdot \vec{b}$.

(A) 5

(B) -1

(C) 1

(D) 0

Question 3. The dot product $\hat{i} \cdot \hat{i}$ is equal to:

(A) 0

(B) 1

(C) $\hat{i}$

(D) $|\hat{i}|$

Question 4. The dot product $\hat{i} \cdot \hat{j}$ is equal to:

(A) 0

(B) 1

(C) $\hat{k}$

(D) $-\hat{k}$

Question 5. For non-zero vectors $\vec{a}$ and $\vec{b}$, $\vec{a} \cdot \vec{b} = 0$ implies that:

(A) $\vec{a}$ and $\vec{b}$ are parallel.

(B) $\vec{a}$ and $\vec{b}$ are perpendicular.

(C) $\vec{a}$ and $\vec{b}$ are equal.

(D) $\vec{a}$ and $\vec{b}$ are collinear.

Question 6. The projection of vector $\vec{a}$ on vector $\vec{b}$ is given by:

(A) $\frac{\vec{a} \cdot \vec{b}}{|\vec{a}|}$

(B) $\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$

(C) $(\vec{a} \cdot \vec{b}) \vec{b}$

(D) $(\vec{a} \cdot \vec{b}) \frac{\vec{b}}{|\vec{b}|}$

Question 7. If $\vec{a}$ and $\vec{b}$ are two vectors, then $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$. This property is called:

(A) Associativity

(B) Distributivity

(C) Commutativity

(D) Identity property

Question 8. The work done by a force $\vec{F}$ causing a displacement $\vec{d}$ is given by:

(A) $\vec{F} + \vec{d}$

(B) $\vec{F} - \vec{d}$

(C) $\vec{F} \cdot \vec{d}$

(D) $\vec{F} \times \vec{d}$

Question 9. The angle $\theta$ between two non-zero vectors $\vec{a}$ and $\vec{b}$ is given by:

(A) $\cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}$

(B) $\sin \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}$

(C) $\tan \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}$

(D) $\theta = \vec{a} \cdot \vec{b}$

Question 10. If $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$ and $\vec{b} = 2\hat{i} - \hat{j} + \hat{k}$, find $\vec{a} \cdot \vec{b}$.

(A) 3

(B) 2

(C) 1

(D) 0

Question 11. If $|\vec{a}| = 3$, $|\vec{b}| = 4$, and the angle between $\vec{a}$ and $\vec{b}$ is $60^\circ$, find $\vec{a} \cdot \vec{b}$.

(A) 6

(B) $6\sqrt{3}$

(C) 12

(D) $12\sqrt{3}$

Question 12. The projection of $\vec{a} = \hat{i} + \hat{j}$ on $\vec{b} = \hat{i}$ is:

(A) 1

(B) $\sqrt{2}$

(C) 0

(D) $\hat{i}$

Question 13. For any vector $\vec{a}$, $\vec{a} \cdot \vec{a}$ is equal to:

(A) 0

(B) $|\vec{a}|$

(C) $|\vec{a}|^2$

(D) $\vec{a}$

Question 14. If $\vec{a}$ and $\vec{b}$ are two vectors, and $k$ is a scalar, then $k(\vec{a} \cdot \vec{b})$ is equal to:

(A) $(k\vec{a}) \cdot \vec{b}$

(B) $\vec{a} \cdot (k\vec{b})$

(C) $(k\vec{a}) \cdot \vec{b}$ and $\vec{a} \cdot (k\vec{b})$ are equal

(D) None of the above

Question 15. If $\vec{a} = \hat{i} - \hat{j} + \hat{k}$ and $\vec{b} = \hat{i} + \hat{j} - \hat{k}$, the angle between $\vec{a}$ and $\vec{b}$ is:

(A) $0^\circ$

(B) $90^\circ$

(C) $60^\circ$

(D) $\cos^{-1}(-\frac{1}{3})$

Question 16. A force $\vec{F} = 3\hat{i} + \hat{j} - 2\hat{k}$ acts on an object, causing a displacement $\vec{d} = \hat{i} + 2\hat{j} + 3\hat{k}$. The work done is:

(A) 1 Joule

(B) -1 Joule

(C) 11 Joule

(D) -11 Joule

Question 17. If $\vec{a}$ is perpendicular to $\vec{b}$, what is $\vec{a} \cdot \vec{b}$?

(A) $|\vec{a}||\vec{b}|$

(B) 1

(C) 0

(D) -1

Question 18. The projection of $\vec{a} = 2\hat{i} - \hat{j} + \hat{k}$ on the vector $\hat{i}$ is:

(A) 2

(B) -1

(C) 1

(D) $\sqrt{6}$

Question 19. If $(\vec{a} + \vec{b}) \cdot (\vec{a} - \vec{b}) = 0$, then:

(A) $\vec{a} = \vec{b}$

(B) $|\vec{a}| = |\vec{b}|$

(C) $\vec{a}$ is parallel to $\vec{b}$

(D) $\vec{a}$ is perpendicular to $\vec{b}$

Question 20. If $\vec{a}$ and $\vec{b}$ are unit vectors such that the angle between them is $30^\circ$, then $\vec{a} \cdot \vec{b}$ is:

(A) 1

(B) 0

(C) $\frac{1}{2}$

(D) $\frac{\sqrt{3}}{2}$

Question 21. The scalar projection of vector $\vec{a}$ on vector $\vec{b}$ is a scalar. The vector projection of $\vec{a}$ on $\vec{b}$ is:

(A) $\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$

(B) $(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}) |\vec{b}|$

(C) $(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}) \hat{b}$

(D) $(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}) \vec{b}$

Question 22. If the angle between two vectors is obtuse, their dot product is:

(A) Positive

(B) Negative

(C) Zero

(D) Cannot be determined

Question 23. For any vector $\vec{v}$, $\vec{v} \cdot \vec{0}$ equals:

(A) $\vec{v}$

(B) $|\vec{v}|$

(C) 0

(D) $\vec{0}$

Question 24. If $\vec{a} = 2\hat{i} + \hat{j}$ and $\vec{b} = \hat{i} + y\hat{j}$, for what value of $y$ are $\vec{a}$ and $\vec{b}$ perpendicular?

(A) 2

(B) -2

(C) 1/2

(D) -1/2

Question 25. The geometrical interpretation of the dot product $\vec{a} \cdot \vec{b}$ is related to:

(A) The area of the parallelogram formed by $\vec{a}$ and $\vec{b}$.

(B) The volume of the parallelepiped formed by $\vec{a}$, $\vec{b}$, and a third vector.

(C) The projection of one vector onto another.

(D) A vector perpendicular to both $\vec{a}$ and $\vec{b}$.

Question 26. If $|\vec{a}| = 2$, $|\vec{b}| = 3$, and $\vec{a} \cdot \vec{b} = 3$, the angle between $\vec{a}$ and $\vec{b}$ is:

(A) $30^\circ$

(B) $45^\circ$

(C) $60^\circ$

(D) $90^\circ$

Question 27. For any two vectors $\vec{a}$ and $\vec{b}$, the inequality $|\vec{a} \cdot \vec{b}| \leq |\vec{a}||\vec{b}|$ is known as:

(A) Triangle inequality

(B) Cauchy-Schwarz inequality

(C) Parallelogram law

(D) Pythagorean theorem

Question 28. The projection of vector $\hat{i} + \hat{j} + \hat{k}$ on the z-axis is:

(A) 1

(B) $\hat{k}$

(C) $\frac{1}{\sqrt{3}}$

(D) $\frac{1}{\sqrt{3}}\hat{k}$

Question 29. A force $\vec{F}$ acts at a point. The force is $\vec{F} = 2\hat{i} + 2\hat{j}$ N. The object moves from point A(1,1) to B(3,4). The work done by the force is:

(A) 6 J

(B) 10 J

(C) 12 J

(D) 14 J

Question 30. If $\vec{a}$ and $\vec{b}$ are parallel, what is $\vec{a} \cdot \vec{b}$?

(A) 0

(B) $|\vec{a}||\vec{b}|$

(C) $\pm |\vec{a}||\vec{b}|$

(D) 1

Question 1. The vector product of two vectors $\vec{a}$ and $\vec{b}$ is a vector defined as $\vec{a} \times \vec{b} = |\vec{a}||\vec{b}|\sin\theta \hat{n}$, where $\hat{n}$ is a unit vector perpendicular to both $\vec{a}$ and $\vec{b}$. The direction of $\hat{n}$ is given by:

(A) Left-hand rule

(B) Right-hand rule

(C) Along $\vec{a}$

(D) Along $\vec{b}$

Question 2. If $\vec{a} = 2\hat{i} + \hat{j}$ and $\vec{b} = \hat{i} - 3\hat{j}$, find $\vec{a} \times \vec{b}$.

(A) $-7\hat{k}$

(B) $7\hat{k}$

(C) $-5\hat{k}$

(D) $5\hat{k}$

Question 3. The cross product $\hat{i} \times \hat{i}$ is equal to:

(A) 0

(B) 1

(C) $\hat{j}$

(D) $\vec{0}$

Question 4. The cross product $\hat{i} \times \hat{j}$ is equal to:

(A) 0

(B) 1

(C) $\hat{k}$

(D) $-\hat{k}$

Question 5. For non-zero vectors $\vec{a}$ and $\vec{b}$, $\vec{a} \times \vec{b} = \vec{0}$ implies that:

(A) $\vec{a}$ and $\vec{b}$ are perpendicular.

(B) $\vec{a}$ and $\vec{b}$ are parallel or collinear.

(C) $\vec{a}$ and $\vec{b}$ are equal.

(D) $|\vec{a}|=|\vec{b}|$.

Question 6. The magnitude of the cross product $|\vec{a} \times \vec{b}|$ represents:

(A) The area of the triangle formed by $\vec{a}$ and $\vec{b}$.

(B) Half the area of the parallelogram formed by $\vec{a}$ and $\vec{b}$.

(C) The area of the parallelogram formed by $\vec{a}$ and $\vec{b}$.

(D) The volume of the parallelepiped formed by $\vec{a}$, $\vec{b}$ and a third vector.

Question 7. For any two vectors $\vec{a}$ and $\vec{b}$, $\vec{a} \times \vec{b}$ is equal to:

(A) $\vec{b} \times \vec{a}$

(B) $-\vec{b} \times \vec{a}$

(C) $\vec{a} \cdot \vec{b}$

(D) $\vec{b} \cdot \vec{a}$

Question 8. If $\vec{F}$ is a force vector and $\vec{r}$ is the position vector from the axis of rotation to the point of application of the force, the torque $\vec{\tau}$ is given by:

(A) $\vec{F} \cdot \vec{r}$

(B) $\vec{r} \cdot \vec{F}$

(C) $\vec{F} \times \vec{r}$

(D) $\vec{r} \times \vec{F}$

Question 9. If $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ and $\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$, then $\vec{a} \times \vec{b}$ is given by the determinant:

(A) $\begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ \hat{i} & \hat{j} & \hat{k} \end{vmatrix}$

(B) $\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$

(C) $\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}$

(D) $a_1b_1 + a_2b_2 + a_3b_3$

Question 10. Find a unit vector perpendicular to both $\vec{a} = \hat{i} + \hat{j}$ and $\vec{b} = \hat{i} - \hat{j}$.

(A) $\hat{k}$

(B) $-\hat{k}$

(C) $\pm \hat{k}$

(D) $\frac{1}{\sqrt{2}}(\hat{i} + \hat{j})$

Question 11. The area of the parallelogram whose adjacent sides are given by vectors $\vec{a} = 3\hat{i} + \hat{j}$ and $\vec{b} = 2\hat{i} - 4\hat{j}$ is:

(A) 14

(B) 7

(C) $\sqrt{14}$

(D) $\sqrt{7}$

Question 12. The area of the triangle with adjacent sides $\vec{a}$ and $\vec{b}$ is given by:

(A) $|\vec{a} \cdot \vec{b}|$

(B) $\frac{1}{2} |\vec{a} \cdot \vec{b}|$

(C) $|\vec{a} \times \vec{b}|$

(D) $\frac{1}{2} |\vec{a} \times \vec{b}|$

Question 13. If $\vec{a}$ and $\vec{b}$ are parallel vectors, what is $\vec{a} \times \vec{b}$?

(A) $|\vec{a}||\vec{b}|$

(B) 0

(C) $\vec{0}$

(D) $|\vec{a}||\vec{b}| \hat{n}$

Question 14. Which property is NOT true for the vector product?

(A) $\vec{a} \times \vec{b} = -\vec{b} \times \vec{a}$

(B) $(k\vec{a}) \times \vec{b} = k(\vec{a} \times \vec{b})$

(C) $\vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}$

(D) $\vec{a} \times \vec{b} = \vec{b} \times \vec{a}$

Question 15. If $|\vec{a}| = 2$, $|\vec{b}| = 3$, and the angle between them is $90^\circ$, find $|\vec{a} \times \vec{b}|$.

(A) 0

(B) 5

(C) 6

(D) 1

Question 16. The area of the triangle with vertices A(1,1,1), B(1,2,3), C(2,3,1) is:

(A) $\frac{1}{2} \sqrt{24}$

(B) $\sqrt{24}$

(C) $\frac{1}{2} \sqrt{21}$

(D) $\sqrt{21}$

Question 17. If $\vec{a} \times \vec{b} = \vec{0}$ and $\vec{a} \cdot \vec{b} = 0$, and $\vec{a}$ is a non-zero vector, then $\vec{b}$ must be:

(A) A non-zero vector parallel to $\vec{a}$.

(B) A non-zero vector perpendicular to $\vec{a}$.

(C) The zero vector.

(D) Any vector.

Question 18. If $\theta$ is the angle between $\vec{a}$ and $\vec{b}$, then $|\vec{a} \times \vec{b}|^2 + (\vec{a} \cdot \vec{b})^2$ equals:

(A) $|\vec{a}|^2 + |\vec{b}|^2$

(B) $(|\vec{a}| + |\vec{b}|)^2$

(C) $|\vec{a}|^2 |\vec{b}|^2$

(D) $(|\vec{a}| - |\vec{b}|)^2$

Question 19. The cross product is useful for finding a vector that is ________ to two given vectors.

(A) Parallel

(B) Collinear

(C) Perpendicular

(D) Coplanar

Question 20. If $\vec{a}$ and $\vec{b}$ are vectors representing diagonals of a parallelogram, its area is given by:

(A) $|\vec{a} \times \vec{b}|$

(B) $\frac{1}{2} |\vec{a} \times \vec{b}|$

(C) $|\vec{a} \cdot \vec{b}|$

(D) $\frac{1}{2} |\vec{a} \cdot \vec{b}|$

Question 21. Given $\vec{a} = \hat{i} + 2\hat{j} - \hat{k}$ and $\vec{b} = 2\hat{i} + 4\hat{j} - 2\hat{k}$. Which statement is true?

(A) $\vec{a} \times \vec{b} = \vec{0}$

(B) $\vec{a} \times \vec{b} \neq \vec{0}$

(C) $\vec{a} \cdot \vec{b} = 0$

(D) $\vec{a} + \vec{b} = \vec{0}$

Question 22. The unit vector perpendicular to the plane containing vectors $\vec{a}$ and $\vec{b}$ is:

(A) $\vec{a} \times \vec{b}$

(B) $\frac{\vec{a} \times \vec{b}}{|\vec{a} \cdot \vec{b}|}$

(C) $\frac{\vec{a} \times \vec{b}}{|\vec{a} \times \vec{b}|}$

(D) $|\vec{a} \times \vec{b}|$

Question 23. If $\vec{a}, \vec{b}, \vec{c}$ are vectors, which of the following is distributive property for cross product?

(A) $\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \times \vec{b}) \times \vec{c}$

(B) $\vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}$

(C) $(\vec{a} + \vec{b}) \times \vec{c} = \vec{c} \times (\vec{a} + \vec{b})$

(D) $\vec{a} \times \vec{a} = |\vec{a}|^2$

Question 24. If $|\vec{a}| = 10$, $|\vec{b}| = 2$, and $|\vec{a} \times \vec{b}| = 12$, then $\vec{a} \cdot \vec{b}$ is:

(A) 16

(B) $\pm 16$

(C) 8

(D) $\pm 8$

Question 25. The area of a triangle with vertices P(1,0,0), Q(0,1,0), R(0,0,1) is:

(A) $\frac{\sqrt{3}}{2}$

(B) $\sqrt{3}$

(C) $\frac{3}{2}$

(D) 3

Question 26. If $\vec{a}$ and $\vec{b}$ are non-zero vectors, and $\vec{a} \times \vec{b} = \vec{0}$, then which of the following must be true?

(A) $\vec{a} = \vec{b}$

(B) $\vec{a}$ is perpendicular to $\vec{b}$

(C) $\vec{a}$ is parallel to $\vec{b}$

(D) $\vec{a}$ and $\vec{b}$ have the same magnitude

Question 27. If the cross product of two non-zero vectors is the zero vector, the angle between them is:

(A) $0^\circ$ or $180^\circ$

(B) $90^\circ$

(C) $45^\circ$

(D) $60^\circ$

Question 28. The magnitude of the torque about the origin produced by a force $\vec{F} = 3\hat{i} - 2\hat{j} + 4\hat{k}$ acting at the point $(1,1,1)$ is:

(A) $\sqrt{26}$

(B) $\sqrt{29}$

(C) $\sqrt{31}$

(D) $\sqrt{33}$

Question 29. The vector product $\vec{a} \times \vec{a}$ is always:

(A) $|\vec{a}|^2$

(B) 0

(C) $\vec{0}$

(D) A unit vector

Question 30. If $\vec{a} = 2\hat{i}$ and $\vec{b} = 3\hat{j}$, then $\vec{a} \times \vec{b}$ is:

(A) 6

(B) $6\hat{i}\hat{j}$

(C) $6\hat{k}$

(D) $\vec{0}$

Question 1. The scalar triple product of three vectors $\vec{a}, \vec{b}, \vec{c}$ is defined as $\vec{a} \cdot (\vec{b} \times \vec{c})$. It is a:

(A) Vector quantity

(B) Scalar quantity

(C) Always positive

(D) Always zero

Question 2. The scalar triple product $[\vec{a}, \vec{b}, \vec{c}]$ is given by the determinant formed by their components. If $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$, $\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$, and $\vec{c} = c_1\hat{i} + c_2\hat{j} + c_3\hat{k}$, the determinant is:

(A) $\begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}$

(B) $\begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix}$

(C) $\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$

(D) $a_1b_1 + a_2b_2 + a_3b_3$

Question 3. The value of $[\hat{i}, \hat{j}, \hat{k}]$ is:

(A) 0

(B) 1

(C) -1

(D) $\hat{i} + \hat{j} + \hat{k}$

Question 4. The scalar triple product $[\vec{a}, \vec{b}, \vec{c}]$ changes sign if:

(A) Any two vectors are interchanged.

(B) All three vectors are cyclically permuted.

(C) One vector is replaced by its negative.

(D) A scalar multiple of one vector is added to another.

Question 5. The volume of the parallelepiped whose adjacent edges are represented by vectors $\vec{a}, \vec{b}, \vec{c}$ is given by:

(A) $\vec{a} \cdot (\vec{b} \times \vec{c})$

(B) $|\vec{a} \cdot (\vec{b} \times \vec{c})|$

(C) $\vec{a} \times (\vec{b} \times \vec{c})$

(D) $|\vec{a} \times (\vec{b} \times \vec{c})|$

Question 6. Three vectors $\vec{a}, \vec{b}, \vec{c}$ are coplanar if and only if:

(A) $\vec{a} \cdot \vec{b} = 0$

(B) $\vec{a} \times \vec{b} = \vec{0}$

(C) $\vec{a} \cdot (\vec{b} \times \vec{c}) = 0$

(D) $\vec{a} + \vec{b} + \vec{c} = \vec{0}$

Question 7. The volume of the tetrahedron with adjacent edges $\vec{a}, \vec{b}, \vec{c}$ is given by:

(A) $|\vec{a} \cdot (\vec{b} \times \vec{c})|$

(B) $\frac{1}{2}|\vec{a} \cdot (\vec{b} \times \vec{c})|$

(C) $\frac{1}{3}|\vec{a} \cdot (\vec{b} \times \vec{c})|$

(D) $\frac{1}{6}|\vec{a} \cdot (\vec{b} \times \vec{c})|$

Question 8. The value of $[\vec{a}, \vec{a}, \vec{b}]$ for any vectors $\vec{a}$ and $\vec{b}$ is:

(A) $|\vec{a}|^2 |\vec{b}|$

(B) 0

(C) $|\vec{a}|^2$

(D) $\vec{0}$

Question 9. If $\vec{a}, \vec{b}, \vec{c}$ are coplanar, their scalar triple product is:

(A) A unit vector

(B) A non-zero scalar

(C) Zero

(D) A vector perpendicular to their plane

Question 10. The value of $\vec{a} \cdot (\vec{b} \times \vec{c})$ is the same as:

(A) $(\vec{a} \cdot \vec{b}) \times \vec{c}$

(B) $(\vec{a} \times \vec{b}) \cdot \vec{c}$

(C) $\vec{a} \times (\vec{b} \cdot \vec{c})$

(D) $\vec{c} \cdot (\vec{a} \times \vec{b})$

Question 11. If $\vec{a} = \hat{i} + 2\hat{j}$, $\vec{b} = \hat{j} + 2\hat{k}$, $\vec{c} = 2\hat{i} + \hat{k}$, find $[\vec{a}, \vec{b}, \vec{c}]$.

(A) -10

(B) 10

(C) -6

(D) 6

Question 12. Three points A(1,0,1), B(1,1,1), C(2,1,1), D(2,0,1) are given. Are these points coplanar?

(A) Yes, because the vectors $\vec{AB}, \vec{AC}, \vec{AD}$ are coplanar.

(B) No, they are not coplanar.

(C) Only if D lies on the line segment AB.

(D) Only if A, B, C are collinear.

Question 13. The scalar triple product $[\vec{a}, \vec{b}, \vec{c}]$ is also denoted by:

(A) $(\vec{a} \cdot \vec{b} \cdot \vec{c})$

(B) $(\vec{a} \times \vec{b} \times \vec{c})$

(C) $[\vec{a} \cdot \vec{b} \cdot \vec{c}]$

(D) $[\vec{a}, \vec{b}, \vec{c}]$

Question 14. If $[\vec{u}, \vec{v}, \vec{w}] \neq 0$, then the vectors $\vec{u}, \vec{v}, \vec{w}$ are:

(A) Coplanar

(B) Non-coplanar

(C) Collinear

(D) Mutually perpendicular

Question 15. The volume of the parallelepiped formed by $\hat{i}, \hat{j}, \hat{k}$ as adjacent edges is:

(A) 0

(B) 1

(C) 3

(D) $\sqrt{3}$

Question 16. If $\vec{a}, \vec{b}, \vec{c}$ are three non-coplanar vectors, which of the following is always non-zero?

(A) $\vec{a} + \vec{b} + \vec{c}$

(B) $\vec{a} \cdot (\vec{b} + \vec{c})$

(C) $\vec{a} \times (\vec{b} - \vec{c})$

(D) $[\vec{a}, \vec{b}, \vec{c}]$

Question 17. The volume of a tetrahedron with vertices P, Q, R, S is given by $\frac{1}{6} |[\vec{PQ}, \vec{PR}, \vec{PS}]|$. If P=(1,1,2), Q=(2,1,3), R=(3,2,1), S=(1,2,3), find the volume.

(A) $\frac{1}{6}$

(B) 1

(C) $\frac{1}{3}$

(D) 2

Question 18. For any permutation $i, j, k$ of $1, 2, 3$, $[\vec{v}_i, \vec{v}_j, \vec{v}_k] = \text{sgn}(i,j,k) [\vec{v}_1, \vec{v}_2, \vec{v}_3]$. This property is related to:

(A) Distributivity

(B) Cyclical permutation

(C) Linear dependence

(D) Commutativity

Question 19. If $\vec{a}, \vec{b}, \vec{c}$ are mutually perpendicular vectors, then $[\vec{a}, \vec{b}, \vec{c}]$ is equal to:

(A) 0

(B) $|\vec{a}||\vec{b}||\vec{c}|$

(C) $\sqrt{|\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2}$

(D) 1

Question 20. If the scalar triple product $[\vec{a}, \vec{b}, \vec{c}]$ is positive, it means:

(A) The vectors form a left-handed system.

(B) The vectors form a right-handed system.

(C) The vectors are coplanar.

(D) The volume is zero.

Question 21. The scalar triple product $[\vec{a} + \vec{b}, \vec{b} + \vec{c}, \vec{c} + \vec{a}]$ is equal to:

(A) 0

(B) $[\vec{a}, \vec{b}, \vec{c}]$

(C) $2[\vec{a}, \vec{b}, \vec{c}]$

(D) $-[\vec{a}, \vec{b}, \vec{c}]$

Question 22. For what value of $\lambda$ are the vectors $\hat{i} - \hat{j} + \hat{k}$, $2\hat{i} + \hat{j} - \hat{k}$, and $\lambda\hat{i} - \hat{j} + \lambda\hat{k}$ coplanar?

(A) 1

(B) 2

(C) -1

(D) Any real number

Question 23. If $\vec{a} = (1, 1, 0)$, $\vec{b} = (1, 0, 1)$, $\vec{c} = (0, 1, 1)$, the volume of the parallelepiped formed by them is:

(A) 0

(B) 1

(C) 2

(D) 3

Question 24. The scalar triple product is zero if:

(A) Two of the vectors are parallel.

(B) All three vectors are mutually perpendicular.

(C) The vectors form a right-handed system.

(D) The vectors form a left-handed system.

Question 25. If the value of the scalar triple product of three vectors is $V$, then the volume of the tetrahedron formed by these vectors as coterminous edges is:

(A) $|V|$

(B) $V/2$

(C) $|V|/6$

(D) $V/3$

Question 26. The property $[\vec{a}, \vec{b}, \vec{c}] = \vec{a} \cdot (\vec{b} \times \vec{c})$ relates the scalar triple product to:

(A) Dot product and vector addition

(B) Cross product and scalar multiplication

(C) Dot product and cross product

(D) Vector addition and cross product

Question 27. If $\vec{a}, \vec{b}, \vec{c}$ are coplanar vectors, which of the following is NOT necessarily true?

(A) $[\vec{a}, \vec{b}, \vec{c}] = 0$

(B) One vector is a linear combination of the other two.

(C) $\vec{a} \cdot (\vec{b} \times \vec{c}) = 0$

(D) $\vec{a} \times (\vec{b} \times \vec{c}) = \vec{0}$

Question 28. The absolute value of the scalar triple product represents a volume. This volume is zero when:

(A) The vectors are linearly independent.

(B) The vectors are linearly dependent.

(C) The vectors are orthonormal.

(D) The vectors form a basis.

Question 29. The value of $[\vec{a}, \vec{b}, \vec{c}]$ is equal to the value of $[\vec{b}, \vec{c}, \vec{a}]$. This is due to:

(A) Interchanging two rows in a determinant changes the sign.

(B) Cyclically permuting rows in a determinant does not change the value.

(C) The dot product is commutative.

(D) The cross product is non-commutative.

Question 30. If the volume of the parallelepiped formed by $\vec{a}, \vec{b}, \vec{c}$ is 5 cubic units, then the volume of the parallelepiped formed by $2\vec{a}, \vec{b}, \vec{c}$ is:

(A) 5

(B) 10

(C) 25

(D) 50

Question 31. For vectors $\vec{a}, \vec{b}, \vec{c}$, the expression $\vec{a} \cdot (\vec{c} \times \vec{b})$ is equal to:

(A) $[\vec{a}, \vec{b}, \vec{c}]$

(B) $-[\vec{a}, \vec{b}, \vec{c}]$

(C) $[\vec{c}, \vec{b}, \vec{a}]$

(D) $\vec{a} \times (\vec{c} \cdot \vec{b})$

Question 32. The vectors $\vec{a} = \hat{i} + \hat{j}$, $\vec{b} = \hat{j} + \hat{k}$, $\vec{c} = \hat{i} + \hat{k}$ are:

(A) Coplanar

(B) Collinear

(C) Non-coplanar

(D) Mutually perpendicular

Question 1. The position vector of a point R that divides the line segment joining points A and B with position vectors $\vec{a}$ and $\vec{b}$ internally in the ratio $m:n$ is given by:

(A) $\frac{n\vec{a} + m\vec{b}}{m+n}$

(B) $\frac{m\vec{a} + n\vec{b}}{m+n}$

(C) $\frac{m\vec{a} - n\vec{b}}{m-n}$

(D) $\frac{n\vec{a} - m\vec{b}}{n-m}$

Question 2. The position vector of the midpoint of the line segment joining points A and B with position vectors $\vec{a}$ and $\vec{b}$ is:

(A) $\frac{\vec{a} + \vec{b}}{2}$

(B) $\frac{\vec{a} - \vec{b}}{2}$

(C) $\vec{a} + \vec{b}$

(D) $|\vec{a} - \vec{b}|$

Question 3. The position vector of a point R that divides the line segment joining points A and B with position vectors $\vec{a}$ and $\vec{b}$ externally in the ratio $m:n$ is given by:

(A) $\frac{n\vec{a} + m\vec{b}}{m+n}$

(B) $\frac{m\vec{a} - n\vec{b}}{m-n}$

(C) $\frac{n\vec{a} - m\vec{b}}{n-m}$

(D) $\frac{m\vec{b} - n\vec{a}}{m-n}$

Question 4. If A, B, C are vertices of a triangle with position vectors $\vec{a}, \vec{b}, \vec{c}$ respectively, the position vector of its centroid G is:

(A) $\frac{\vec{a} + \vec{b} + \vec{c}}{3}$

(B) $\vec{a} + \vec{b} + \vec{c}$

(C) $\frac{\vec{a} + \vec{b}}{2}$

(D) $\frac{\vec{a} + \vec{b} + \vec{c}}{2}$

Question 5. The position vectors of points P and Q are $\hat{i} + 2\hat{j} - \hat{k}$ and $-\hat{i} + \hat{j} + \hat{k}$. Find the position vector of the midpoint of PQ.

(A) $\hat{j}$

(B) $\hat{j} + \hat{k}$

(C) $\hat{i} + \frac{3}{2}\hat{j}$

(D) $\hat{j} + \frac{1}{2}\hat{k}$

Question 6. The position vectors of A and B are $\vec{a}$ and $\vec{b}$. A point C divides the line segment AB internally in the ratio $1:2$. What is the position vector of C?

(A) $\frac{\vec{a} + 2\vec{b}}{3}$

(B) $\frac{2\vec{a} + \vec{b}}{3}$

(C) $\frac{\vec{a} + \vec{b}}{3}$

(D) $\vec{a} + 2\vec{b}$

Question 7. If a point R divides the line segment AB in the ratio $1:1$, it is the same as finding the:

(A) Centroid

(B) External division point

(C) Midpoint

(D) Terminal point

Question 8. The position vectors of the vertices of a triangle are $\hat{i}$, $\hat{j}$, $\hat{k}$. The position vector of its centroid is:

(A) $\hat{i} + \hat{j} + \hat{k}$

(B) $\frac{1}{3}(\hat{i} + \hat{j} + \hat{k})$

(C) $\frac{1}{2}(\hat{i} + \hat{j} + \hat{k})$

(D) $\vec{0}$

Question 9. If the position vectors of points A and B are $\vec{a}$ and $\vec{b}$, and point C divides AB externally in the ratio $2:1$, the position vector of C is:

(A) $2\vec{b} - \vec{a}$

(B) $2\vec{a} - \vec{b}$

(C) $\vec{b} - 2\vec{a}$

(D) $\vec{a} - 2\vec{b}$

Question 10. Point R has position vector $\vec{r}$. If R divides the line segment PQ, with position vectors $\vec{p}$ and $\vec{q}$, such that $\vec{r} = \frac{3\vec{q} + 2\vec{p}}{5}$, R divides PQ in the ratio:

(A) 3:2 internally

(B) 2:3 internally

(C) 3:2 externally

(D) 2:3 externally

Question 11. The position vectors of the vertices of a tetrahedron are $\vec{a}, \vec{b}, \vec{c}, \vec{d}$. The position vector of its centroid is:

(A) $\frac{\vec{a} + \vec{b} + \vec{c} + \vec{d}}{4}$

(B) $\frac{\vec{a} + \vec{b} + \vec{c}}{3}$

(C) $\vec{a} + \vec{b} + \vec{c} + \vec{d}$

(D) $\frac{1}{4}(\vec{a} + \vec{b} + \vec{c})$

Question 12. If point C divides AB in the ratio $m:n$, and A, B, C are collinear, the section formula is based on which geometric principle?

(A) Pythagorean Theorem

(B) Similarity of triangles

(C) Vector addition

(D) Dot product

Question 13. The position vectors of A and B are $2\hat{i} - \hat{j} + \hat{k}$ and $\hat{i} + \hat{j} + 2\hat{k}$. Find the position vector of the point that divides AB internally in the ratio $2:3$.

(A) $\frac{8\hat{i} - \hat{j} + 7\hat{k}}{5}$

(B) $\frac{8\hat{i} + \hat{j} - 7\hat{k}}{5}$

(C) $\frac{8\hat{i} + \hat{j} + 7\hat{k}}{5}$

(D) $\frac{4\hat{i} - \hat{j} + 5\hat{k}}{5}$

Question 14. If a point P divides AB externally in the ratio $n:m$, the formula is equivalent to dividing it internally in the ratio:

(A) $n:m$

(B) $m:n$

(C) $-n:m$ or $n:-m$

(D) $1:1$

Question 15. If the position vector of point R is $\vec{r} = \frac{2\vec{a} - 5\vec{b}}{2-5}$, R divides the line segment joining points A (pos. vector $\vec{a}$) and B (pos. vector $\vec{b}$) in the ratio:

(A) 2:5 internally

(B) 5:2 internally

(C) 2:5 externally

(D) 5:2 externally

Question 16. The position vector of the centroid of a triangle is the average of the position vectors of its vertices. This point is the intersection of the triangle's:

(A) Altitudes

(B) Medians

(C) Angle bisectors

(D) Perpendicular bisectors

Question 17. If the origin is O and point P divides AB internally in the ratio $m:n$, then $\vec{OP}$ is given by:

(A) $\frac{n\vec{OA} + m\vec{OB}}{m+n}$

(B) $\frac{m\vec{OA} + n\vec{OB}}{m+n}$

(C) $\frac{n\vec{OA} - m\vec{OB}}{n-m}$

(D) $\frac{m\vec{OB} - n\vec{OA}}{m-n}$

Question 18. Point M is the midpoint of line segment PQ. If the position vector of P is $\vec{p}$ and the position vector of M is $\vec{m}$, what is the position vector of Q?

(A) $2\vec{m} - \vec{p}$

(B) $2\vec{p} - \vec{m}$

(C) $\vec{m} - \vec{p}$

(D) $\vec{m} + \vec{p}$

Question 19. The position vectors of A, B, C are $\vec{a}, \vec{b}, \vec{c}$. If G is the centroid of triangle ABC, then $\vec{GA} + \vec{GB} + \vec{GC}$ equals:

(A) $\vec{a} + \vec{b} + \vec{c}$

(B) $\vec{0}$

(C) $3\vec{OG}$ (where O is origin)

(D) $\vec{a} + \vec{b} + \vec{c} - 3\vec{OG}$

Question 20. If point R divides the line segment joining points A and B in the ratio $m:n$, and R lies on the line segment between A and B, it is:

(A) External division

(B) Internal division

(C) Midpoint division

(D) Cannot be determined

Question 21. Points A, B, C are collinear. If C divides AB externally in the ratio $m:n$, with $m \neq n$, then C lies on the line AB but:

(A) Between A and B

(B) Outside the segment AB

(C) At A

(D) At B

Question 22. The position vector of the centroid of the tetrahedron with vertices (1,0,0), (0,1,0), (0,0,1), (1,1,1) is:

(A) $(\frac{1}{2}, \frac{1}{2}, \frac{1}{2})$

(B) $(\frac{2}{3}, \frac{2}{3}, \frac{2}{3})$

(C) $(\frac{1}{4}, \frac{1}{4}, \frac{1}{4})$

(D) $(\frac{3}{4}, \frac{3}{4}, \frac{3}{4})$

Question 1. In a 3D coordinate system, the yz-plane is defined by the equation:

(A) $x=0$

(B) $y=0$

(C) $z=0$

(D) $x+y+z=0$

Question 2. The distance between points P(1, -2, 3) and Q(3, -1, -1) is:

(A) $\sqrt{20}$

(B) $\sqrt{21}$

(C) $\sqrt{22}$

(D) $\sqrt{23}$

Question 3. If a line makes angles $\alpha, \beta, \gamma$ with the positive x, y, z axes, respectively, then its direction cosines are:

(A) $\sin \alpha, \sin \beta, \sin \gamma$

(B) $\cos \alpha, \cos \beta, \cos \gamma$

(C) $\tan \alpha, \tan \beta, \tan \gamma$

(D) $\sec \alpha, \sec \beta, \sec \gamma$

Question 4. The sum of the squares of the direction cosines of any line is always equal to:

(A) 0

(B) 1

(C) -1

(D) 3

Question 5. If $a, b, c$ are direction ratios of a line, its direction cosines $\ell, m, n$ are given by:

(A) $\ell = ka, m = kb, n = kc$ for some constant $k$.

(B) $\ell = \frac{a}{a+b+c}, m = \frac{b}{a+b+c}, n = \frac{c}{a+b+c}$

(C) $\ell = \frac{a}{\sqrt{a^2+b^2+c^2}}, m = \frac{b}{\sqrt{a^2+b^2+c^2}}, n = \frac{c}{\sqrt{a^2+b^2+c^2}}$

(D) $\ell = \sqrt{a}, m = \sqrt{b}, n = \sqrt{c}$

Question 6. The vector equation of a line passing through a point with position vector $\vec{a}$ and parallel to vector $\vec{b}$ is:

(A) $\vec{r} = \vec{a} + \vec{b}$

(B) $\vec{r} = \vec{a} + \lambda \vec{b}$

(C) $\vec{r} = \lambda (\vec{a} + \vec{b})$

(D) $\vec{r} = \lambda \vec{a} + \vec{b}$

Question 7. The Cartesian equation of a line passing through $(x_1, y_1, z_1)$ with direction ratios $a, b, c$ is:

(A) $\frac{x}{a} = \frac{y}{b} = \frac{z}{c}$

(B) $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$

(C) $\frac{x_1-x}{a} = \frac{y_1-y}{b} = \frac{z_1-z}{c}$

(D) $a(x-x_1) + b(y-y_1) + c(z-z_1) = 0$

Question 8. The vector equation of a line passing through two points with position vectors $\vec{a}$ and $\vec{b}$ is:

(A) $\vec{r} = \vec{a} + \lambda \vec{b}$

(B) $\vec{r} = \vec{a} + \lambda (\vec{b} - \vec{a})$

(C) $\vec{r} = \lambda \vec{a} + (1-\lambda) \vec{b}$

(D) $\vec{r} = (1-\lambda) \vec{a} + \lambda \vec{b}$

Question 9. The Cartesian equation of a line passing through $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ is:

(A) $\frac{x}{x_2-x_1} = \frac{y}{y_2-y_1} = \frac{z}{z_2-z_1}$

(B) $\frac{x-x_1}{x_2-x_1} = \frac{y-y_1}{y_2-y_1} = \frac{z-z_1}{z_2-z_1}$

(C) $\frac{x-x_1}{x_1-x_2} = \frac{y-y_1}{y_1-y_2} = \frac{z-z_1}{z_1-z_2}$

(D) Both (B) and (C)

Question 10. The direction cosines of the z-axis are:

(A) (1, 0, 0)

(B) (0, 1, 0)

(C) (0, 0, 1)

(D) (1, 1, 1)

Question 11. If a line has direction ratios $1, -2, 2$, its direction cosines are:

(A) $(\frac{1}{3}, -\frac{2}{3}, \frac{2}{3})$

(B) $(1, -2, 2)$

(C) $(\frac{1}{\sqrt{6}}, -\frac{2}{\sqrt{6}}, \frac{2}{\sqrt{6}})$

(D) $(\frac{1}{9}, \frac{4}{9}, \frac{4}{9})$

Question 12. The vector equation of the line $\frac{x-1}{2} = \frac{y+3}{1} = \frac{z-4}{-1}$ is:

(A) $\vec{r} = (2\hat{i} + \hat{j} - \hat{k}) + \lambda (\hat{i} - 3\hat{j} + 4\hat{k})$

(B) $\vec{r} = (\hat{i} - 3\hat{j} + 4\hat{k}) + \lambda (2\hat{i} + \hat{j} - \hat{k})$

(C) $\vec{r} = (\hat{i} + 3\hat{j} - 4\hat{k}) + \lambda (2\hat{i} + \hat{j} - \hat{k})$

(D) $\vec{r} = (2\hat{i} + \hat{j} - \hat{k}) + \lambda (\hat{i} + 3\hat{j} - 4\hat{k})$

Question 13. The Cartesian equation of the line $\vec{r} = (2\hat{i} - \hat{j} + 3\hat{k}) + \lambda (\hat{i} + 4\hat{j} - 2\hat{k})$ is:

(A) $\frac{x-2}{1} = \frac{y+1}{4} = \frac{z-3}{-2}$

(B) $\frac{x-1}{2} = \frac{y-4}{-1} = \frac{z+2}{3}$

(C) $\frac{x+2}{1} = \frac{y-1}{4} = \frac{z+3}{-2}$

(D) $\frac{x+1}{2} = \frac{y+4}{-1} = \frac{z-2}{3}$

Question 14. The coordinates of a point on the line $\frac{x-2}{3} = \frac{y+1}{4} = \frac{z-5}{2}$ for $\lambda=1$ are:

(A) $(5, 3, 7)$

(B) $(3, 4, 2)$

(C) $(1, -5, 3)$

(D) $(-1, -5, 3)$

Question 15. The direction ratios of the line passing through points $(1, 2, -3)$ and $(3, 5, 1)$ are proportional to:

(A) $(2, 3, 4)$

(B) $(-2, -3, -4)$

(C) $(4, 7, -2)$

(D) Both (A) and (B)

Question 16. If the direction cosines of a line are $(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}})$, the angles it makes with the coordinate axes are:

(A) $45^\circ, 45^\circ, 45^\circ$

(B) $60^\circ, 60^\circ, 60^\circ$

(C) $30^\circ, 30^\circ, 30^\circ$

(D) $\cos^{-1}(\frac{1}{\sqrt{3}}), \cos^{-1}(\frac{1}{\sqrt{3}}), \cos^{-1}(\frac{1}{\sqrt{3}})$

Question 17. The projection of a line segment joining points P and Q on a coordinate axis is the difference between the corresponding coordinates of P and Q. The projection of PQ on the x-axis, where P is $(x_1, y_1, z_1)$ and Q is $(x_2, y_2, z_2)$, is:

(A) $x_2 - x_1$

(B) $|x_2 - x_1|$

(C) $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$

(D) $x_1 + x_2$

Question 18. The distance of the point $(2, 3, 4)$ from the origin is:

(A) $\sqrt{9}$

(B) $\sqrt{13}$

(C) $\sqrt{29}$

(D) $\sqrt{52}$

Question 19. The point that divides the line segment joining A(1,-1,2) and B(2,1,-1) internally in the ratio 2:1 has coordinates:

(A) $(1, \frac{1}{3}, \frac{1}{3})$

(B) $(\frac{4}{3}, \frac{1}{3}, 1)$

(C) $(\frac{5}{3}, \frac{1}{3}, 0)$

(D) $(1, 0, \frac{1}{3})$

Question 20. The direction ratios of a line are proportional to its direction cosines. If the direction cosines are $(\ell, m, n)$, the direction ratios can be $(k\ell, km, kn)$ for any non-zero scalar $k$. Which of the following cannot be direction cosines of a line?

(A) $(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0)$

(B) $(1, 0, 0)$

(C) $(\frac{1}{2}, \frac{1}{2}, \frac{1}{2})$

(D) $(\frac{1}{\sqrt{6}}, \frac{2}{\sqrt{6}}, \frac{1}{\sqrt{6}})$

Question 21. The line $\vec{r} = (2\hat{i} + \hat{j}) + \lambda (-3\hat{i} + 5\hat{j} + \hat{k})$ passes through the point:

(A) $(-3, 5, 1)$

(B) $(2, 1, 0)$

(C) $(2, 1, -1)$

(D) $(-3, 5, 0)$

Question 22. The direction vector of the line $\frac{x-1}{5} = \frac{y+2}{-2} = \frac{z}{3}$ is:

(A) $\hat{i} - 2\hat{j} + 0\hat{k}$

(B) $5\hat{i} - 2\hat{j} + 3\hat{k}$

(C) $5\hat{i} + 2\hat{j} + 3\hat{k}$

(D) $\hat{i} - 2\hat{j} + 3\hat{k}$

Question 23. If a line makes equal angles with the x, y, and z axes, its direction cosines are:

(A) $(\pm \frac{1}{\sqrt{3}}, \pm \frac{1}{\sqrt{3}}, \pm \frac{1}{\sqrt{3}})$ (any combination of signs)

(B) $(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}})$ or $(-\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}})$

(C) $(1, 1, 1)$

(D) $(1/\sqrt{3}, 1/\sqrt{3}, 1/\sqrt{3})$ only positive values

Question 24. The point $(x, y, z)$ lies on the line $\vec{r} = \vec{a} + \lambda \vec{b}$. This means the vector $\vec{r} - \vec{a}$ is parallel to $\vec{b}$. The vector $\vec{r} - \vec{a}$ represents:

(A) The position vector of a point on the line.

(B) A vector from the origin to a point on the line.

(C) A vector from point $\vec{a}$ to point $\vec{r}$ on the line.

(D) The direction vector of the line.

Question 1. The equation of a plane in vector form, where $\hat{n}$ is a unit normal vector to the plane and $d$ is the perpendicular distance from the origin to the plane, is:

(A) $\vec{r} \cdot \hat{n} = d$

(B) $\vec{r} \times \hat{n} = d$

(C) $\vec{r} + \hat{n} = d$

(D) $\vec{r} \cdot \vec{n} = d |\vec{n}|$

Question 2. The Cartesian equation of a plane is typically written as $Ax + By + Cz + D = 0$. The vector $(A, B, C)$ is:

(A) A vector parallel to the plane.

(B) A normal vector to the plane.

(C) A point on the plane.

(D) The direction vector of a line in the plane.

Question 3. The equation of a plane passing through a point with position vector $\vec{a}$ and perpendicular to vector $\vec{n}$ is:

(A) $\vec{r} \cdot \vec{n} = \vec{a} \cdot \vec{n}$

(B) $\vec{r} \cdot \vec{a} = \vec{a} \cdot \vec{n}$

(C) $\vec{r} \times \vec{n} = \vec{a} \times \vec{n}$

(D) $(\vec{r} - \vec{a}) \times \vec{n} = \vec{0}$

Question 4. The intercept form of the equation of a plane is $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$. Here, $a, b, c$ represent:

(A) The distances from the origin to the plane along the axes.

(B) The intercepts on the x, y, and z axes respectively.

(C) The coordinates of a point on the plane.

(D) The direction ratios of the normal to the plane.

Question 5. The vector equation $\vec{r} \cdot (2\hat{i} + 3\hat{j} - \hat{k}) = 5$ represents a plane. What are the direction ratios of the normal to this plane?

(A) $(2, 3, 1)$

(B) $(2, 3, -1)$

(C) $(5, 0, 0)$

(D) $(2/5, 3/5, -1/5)$

Question 6. Convert the Cartesian equation $2x - y + z = 4$ into vector form (normal form). The normal vector $\hat{n}$ is:

(A) $2\hat{i} - \hat{j} + \hat{k}$

(B) $\frac{1}{\sqrt{6}}(2\hat{i} - \hat{j} + \hat{k})$

(C) $\frac{1}{\sqrt{6}}(2\hat{i} - \hat{j} - \hat{k})$

(D) $2\hat{i} - \hat{j} + \hat{k}$ and $d=4$

Question 7. The equation of a plane passing through the origin is of the form $Ax + By + Cz = 0$. What is the distance of this plane from the origin?

(A) $D$

(B) $\frac{|D|}{\sqrt{A^2+B^2+C^2}}$

(C) 0

(D) Cannot be determined

Question 8. The equation of a plane passing through three non-collinear points A, B, C with position vectors $\vec{a}, \vec{b}, \vec{c}$ is given by $\vec{r} \cdot \vec{n} = \vec{a} \cdot \vec{n}$, where $\vec{n}$ is a normal to the plane. A suitable normal vector $\vec{n}$ can be:

(A) $\vec{a} \times \vec{b}$

(B) $(\vec{b} - \vec{a}) \times (\vec{c} - \vec{a})$

(C) $\vec{a} + \vec{b} + \vec{c}$

(D) $(\vec{a} + \vec{b}) \times (\vec{b} + \vec{c})$

Question 9. The equation of any plane passing through the intersection of planes $P_1: \vec{r} \cdot \vec{n}_1 = d_1$ and $P_2: \vec{r} \cdot \vec{n}_2 = d_2$ is given by:

(A) $\vec{r} \cdot (\vec{n}_1 + \lambda \vec{n}_2) = d_1 + d_2$

(B) $\vec{r} \cdot (\vec{n}_1 + \lambda \vec{n}_2) = d_1 + \lambda d_2$

(C) $\vec{r} \cdot (\lambda \vec{n}_1 + \vec{n}_2) = \lambda d_1 + d_2$

(D) $\vec{r} \cdot (\vec{n}_1 - \lambda \vec{n}_2) = d_1 - \lambda d_2$

Question 10. The equation of the plane passing through $(1, 2, 3)$ and perpendicular to the vector $3\hat{i} - \hat{j} + 2\hat{k}$ is:

(A) $3(x-1) - (y-2) + 2(z-3) = 0$

(B) $3x - y + 2z + 7 = 0$

(C) $x + 2y + 3z = 0$

(D) $3x - y + 2z = 0$

Question 11. The Cartesian equation of the plane $\vec{r} \cdot (\hat{i} + 2\hat{j} - 3\hat{k}) = 7$ is:

(A) $x + 2y - 3z = 7$

(B) $x + 2y - 3z - 7 = 0$

(C) $x + 2y - 3z = 0$

(D) $x + 2y - 3z + 7 = 0$

Question 12. The intercepts on the coordinate axes made by the plane $2x + 3y - 4z = 12$ are:

(A) 2, 3, -4

(B) 12, 12, 12

(C) 6, 4, -3

(D) $\frac{1}{6}, \frac{1}{4}, -\frac{1}{3}$

Question 13. A plane is uniquely determined by:

(A) Two distinct points.

(B) A point and a vector parallel to the plane.

(C) Three collinear points.

(D) Three non-collinear points.

Question 14. The equation of the plane passing through $(1,0,0), (0,1,0), (0,0,1)$ is:

(A) $x+y+z = 1$

(B) $x+y+z = 0$

(C) $x+y+z = 3$

(D) $x+y+z = -1$

Question 15. The equation of the plane passing through the intersection of $x+y+z=6$ and $2x+3y+4z=5$ and through the point $(1,1,1)$ is of the form $x+y+z-6 + \lambda(2x+3y+4z-5)=0$. The value of $\lambda$ is:

(A) 1

(B) -1

(C) 0

(D) 2

Question 16. The normal vector to the plane $5x - 2y + 7z - 11 = 0$ is:

(A) $(5, -2, 7)$

(B) $(5, -2, 7, -11)$

(C) $(5, -2, 7, 11)$

(D) $(-5, 2, -7)$

Question 17. The distance of the plane $2x - y + 2z = 6$ from the origin is:

(A) 6

(B) $\frac{|6|}{\sqrt{2^2+(-1)^2+2^2}} = \frac{6}{\sqrt{9}} = 2$

(C) $\frac{6}{\sqrt{4+1+4}} = 2$

(D) 3

Question 18. The equation of the plane passing through the point $(1, -1, 2)$ and parallel to the plane $2x + 3y - z = 5$ is:

(A) $2x + 3y - z = 5$

(B) $2x + 3y - z = -3$

(C) $2x + 3y - z = -5$

(D) $2x + 3y - z = 2+3-2 = 3$

Question 19. The plane $\vec{r} \cdot \hat{n} = d$ is perpendicular to the vector $\vec{n}$. If $\vec{n}$ is not a unit vector, the distance from the origin is:

(A) $d$

(B) $d/|\vec{n}|$

(C) $d \cdot |\vec{n}|$

(D) $|d|/|\vec{n}|$

Question 20. Which of the following represents a plane passing through the origin?

(A) $3x - 4y + 5z = 1$

(B) $x+y+z=0$

(C) $\vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 5$

(D) $\frac{x}{1} + \frac{y}{2} + \frac{z}{3} = 1$

Question 21. The vector equation of the plane $x+y+z=3$ is:

(A) $\vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 3$

(B) $\vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = \sqrt{3}$

(C) $\vec{r} \cdot (\frac{\hat{i}}{\sqrt{3}} + \frac{\hat{j}}{\sqrt{3}} + \frac{\hat{k}}{\sqrt{3}}) = 3$

(D) $\vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 1$

Question 22. The direction cosines of the normal to the plane $x - 2y + 2z + 1 = 0$ are:

(A) $(1, -2, 2)$

(B) $(-1, 2, -2)$

(C) $(\frac{1}{3}, -\frac{2}{3}, \frac{2}{3})$

(D) $(\frac{-1}{3}, \frac{2}{3}, -\frac{2}{3})$

Question 23. The equation of the plane passing through $(1,0,0), (0,1,0)$ and $(0,0,0)$ is:

(A) $x+y+z=0$

(B) $z=0$

(C) $x=0$

(D) $y=0$

Question 24. The general equation of a plane contains terms up to degree:

(A) 0

(B) 1

(C) 2

(D) 3

Question 1. The angle $\theta$ between two lines with direction ratios $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ is given by $\cos \theta =$:

(A) $\frac{|a_1a_2 + b_1b_2 + c_1c_2|}{\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}}$

(B) $\frac{|a_1a_2 + b_1b_2 + c_1c_2|}{(a_1^2+b_1^2+c_1^2)(a_2^2+b_2^2+c_2^2)}$

(C) $\frac{\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}}{|a_1a_2 + b_1b_2 + c_1c_2|}$

(D) $a_1a_2 + b_1b_2 + c_1c_2$

Question 2. Two lines with direction ratios $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ are perpendicular if:

(A) $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$

(B) $a_1a_2 + b_1b_2 + c_1c_2 = 0$

(C) $a_1a_2 + b_1b_2 + c_1c_2 = 1$

(D) $a_1+a_2 = b_1+b_2 = c_1+c_2$

Question 3. The angle $\theta$ between two planes $A_1x + B_1y + C_1z + D_1 = 0$ and $A_2x + B_2y + C_2z + D_2 = 0$ is the angle between their normal vectors $(A_1, B_1, C_1)$ and $(A_2, B_2, C_2)$. $\cos \theta =$:

(A) $\frac{|A_1A_2 + B_1B_2 + C_1C_2|}{\sqrt{A_1^2+B_1^2+C_1^2} \sqrt{A_2^2+B_2^2+C_2^2}}$

(B) $\frac{|A_1A_2 + B_1B_2 + C_1C_2|}{(A_1^2+B_1^2+C_1^2)(A_2^2+B_2^2+C_2^2)}$

(C) $\frac{|D_1D_2|}{\sqrt{A_1^2+B_1^2+C_1^2} \sqrt{A_2^2+B_2^2+C_2^2}}$

(D) $A_1A_2 + B_1B_2 + C_1C_2$

Question 4. Two planes $A_1x + B_1y + C_1z + D_1 = 0$ and $A_2x + B_2y + C_2z + D_2 = 0$ are parallel if:

(A) $A_1A_2 + B_1B_2 + C_1C_2 = 0$

(B) $\frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2}$

(C) $\frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2} = \frac{D_1}{D_2}$

(D) $\frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2} \neq \frac{D_1}{D_2}$

Question 5. The angle $\phi$ between a line with direction ratios $(a, b, c)$ and a plane $Ax + By + Cz + D = 0$ is given by $\sin \phi =$:

(A) $\frac{|aA + bB + cC|}{\sqrt{a^2+b^2+c^2} \sqrt{A^2+B^2+C^2}}$

(B) $\frac{|aA + bB + cC|}{(a^2+b^2+c^2)(A^2+B^2+C^2)}$

(C) $\frac{|aA + bB + cC|}{A^2+B^2+C^2}$

(D) $\frac{\sqrt{a^2+b^2+c^2}}{\sqrt{A^2+B^2+C^2}}$

Question 6. A line with direction vector $\vec{v}$ is parallel to a plane with normal vector $\vec{n}$ if:

(A) $\vec{v} \cdot \vec{n} = 0$

(B) $\vec{v} \times \vec{n} = \vec{0}$

(C) $\vec{v} = k\vec{n}$ for some scalar $k$.

(D) $\vec{v} \cdot \vec{n} \neq 0$

Question 7. A line with direction vector $\vec{v}$ is perpendicular to a plane with normal vector $\vec{n}$ if:

(A) $\vec{v} \cdot \vec{n} = 0$

(B) $\vec{v} \times \vec{n} = \vec{0}$

(C) $\vec{v} = k\vec{n}$ for some scalar $k$.

(D) Both (B) and (C)

Question 8. The angle between the lines $\vec{r} = \vec{a}_1 + \lambda \vec{b}_1$ and $\vec{r} = \vec{a}_2 + \mu \vec{b}_2$ is the angle between their direction vectors $\vec{b}_1$ and $\vec{b}_2$. This angle is given by $\cos \theta =$:

(A) $\frac{|\vec{b}_1 \cdot \vec{b}_2|}{|\vec{a}_1||\vec{a}_2|}$

(B) $\frac{|\vec{b}_1 \cdot \vec{b}_2|}{|\vec{b}_1||\vec{b}_2|}$

(C) $\frac{|\vec{a}_1 \cdot \vec{a}_2|}{|\vec{a}_1||\vec{a}_2|}$

(D) $\frac{|\vec{a}_1 \times \vec{a}_2|}{|\vec{a}_1||\vec{a}_2|}$

Question 9. The angle between the planes $\vec{r} \cdot \vec{n}_1 = d_1$ and $\vec{r} \cdot \vec{n}_2 = d_2$ is the angle between their normal vectors $\vec{n}_1$ and $\vec{n}_2$. This angle $\theta$ is given by $\cos \theta =$:

(A) $\frac{|\vec{n}_1 \cdot \vec{n}_2|}{|\vec{n}_1||\vec{n}_2|}$

(B) $\frac{|\vec{n}_1 \times \vec{n}_2|}{|\vec{n}_1||\vec{n}_2|}$

(C) $\frac{|d_1 - d_2|}{|\vec{n}_1||\vec{n}_2|}$

(D) $\frac{|d_1 + d_2|}{|\vec{n}_1||\vec{n}_2|}$

Question 10. The angle between the line $\frac{x-1}{2} = \frac{y+1}{3} = \frac{z-4}{6}$ and the plane $3x + 2y - 2z = 4$ is $\phi$. $\sin \phi =$:

(A) $\frac{|2(3) + 3(2) + 6(-2)|}{\sqrt{2^2+3^2+6^2} \sqrt{3^2+2^2+(-2)^2}}$

(B) $\frac{|2(3) + 3(2) + 6(-2)|}{\sqrt{2^2+3^2+6^2}}$

(C) $\frac{|2(3) + 3(2) + 6(-2)|}{\sqrt{3^2+2^2+(-2)^2}}$

(D) $\frac{|2(3) + 3(2) + 6(2)|}{\sqrt{2^2+3^2+6^2} \sqrt{3^2+2^2+2^2}}$

Question 11. Two lines $\frac{x-x_1}{a_1} = \frac{y-y_1}{b_1} = \frac{z-z_1}{c_1}$ and $\frac{x-x_2}{a_2} = \frac{y-y_2}{b_2} = \frac{z-z_2}{c_2}$ are parallel if:

(A) $a_1a_2 + b_1b_2 + c_1c_2 = 0$

(B) $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$

(C) They are skew lines.

(D) They intersect at a point.

Question 12. The angle between the lines $\vec{r} = \lambda (\hat{i} + 2\hat{j} + 3\hat{k})$ and $\vec{r} = \mu (2\hat{i} - \hat{j} + 0\hat{k})$ is $\theta$. $\cos \theta =$:

(A) $\frac{|1(2) + 2(-1) + 3(0)|}{\sqrt{1^2+2^2+3^2} \sqrt{2^2+(-1)^2+0^2}} = \frac{0}{\sqrt{14}\sqrt{5}} = 0$

(B) $\frac{0}{\sqrt{14}\sqrt{5}}$

(C) $\frac{|1(2) - 2(1)|}{\sqrt{14}\sqrt{5}}$

(D) $\frac{0}{\sqrt{19}\sqrt{5}}$

Question 13. Two planes $A_1x + B_1y + C_1z + D_1 = 0$ and $A_2x + B_2y + C_2z + D_2 = 0$ are perpendicular if:

(A) $A_1A_2 + B_1B_2 + C_1C_2 = 0$

(B) $\frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2}$

(C) They are coincident.

(D) They intersect at a line.

Question 14. The angle between the plane $x+y+z=1$ and the plane $x=0$ is $\theta$. The normal vectors are $\vec{n}_1 = \hat{i} + \hat{j} + \hat{k}$ and $\vec{n}_2 = \hat{i}$. $\cos \theta =$:

(A) $\frac{|\hat{i} \cdot (\hat{i} + \hat{j} + \hat{k})|}{|\hat{i}| |\hat{i} + \hat{j} + \hat{k}|} = \frac{|1|}{\sqrt{1}\sqrt{3}} = \frac{1}{\sqrt{3}}$

(B) $\frac{1}{3}$

(C) $\frac{1}{\sqrt{2}}$

(D) $\frac{1}{2}$

Question 15. A line is parallel to a plane if its direction vector is perpendicular to the plane's normal vector. The line $\frac{x}{1} = \frac{y}{2} = \frac{z}{3}$ is parallel to the plane $x + 2y - z = 5$. Is this true?

(A) Yes, because $(1)(1) + (2)(2) + (3)(-1) = 1+4-3 = 2 \neq 0$.

(B) Yes, because $(1)(1) + (2)(2) + (3)(-1) = 1+4-3 = 2 \neq 0$. Wait, parallel means dot product is zero. The dot product is not zero. So they are not parallel?

(C) No, because the direction vector $(1,2,3)$ is not perpendicular to the normal vector $(1,2,-1)$, since $(1)(1) + (2)(2) + (3)(-1) = 1+4-3=2 \neq 0$.

(D) Yes, because their direction ratios are proportional.

Question 16. The angle between the line $\vec{r} = (1-\lambda)\hat{i} + \lambda\hat{j}$ and the plane $\vec{r} \cdot (\hat{i} + \hat{k}) = 5$ is $\phi$. The direction vector of the line is $-\hat{i} + \hat{j} + 0\hat{k}$. The normal vector of the plane is $\hat{i} + 0\hat{j} + \hat{k}$. $\sin \phi =$:

(A) $\frac{|(-\hat{i} + \hat{j}) \cdot (\hat{i} + \hat{k})|}{|-\hat{i} + \hat{j}| |\hat{i} + \hat{k}|} = \frac{|-1|}{\sqrt{2}\sqrt{2}} = \frac{1}{2}$

(B) $\frac{1}{\sqrt{2}}$

(C) 0

(D) 1

Question 17. The condition for two lines with direction cosines $(\ell_1, m_1, n_1)$ and $(\ell_2, m_2, n_2)$ to be parallel is:

(A) $\ell_1\ell_2 + m_1m_2 + n_1n_2 = 0$

(B) $\ell_1\ell_2 + m_1m_2 + n_1n_2 = 1$

(C) $\frac{\ell_1}{\ell_2} = \frac{m_1}{m_2} = \frac{n_1}{n_2}$

(D) $\ell_1 = \ell_2, m_1 = m_2, n_1 = n_2$

Question 18. The angle between the x-axis and the plane $x+y+z=1$ is:

(A) $0^\circ$

(B) $90^\circ$

(C) $\sin^{-1}(\frac{1}{\sqrt{3}})$

(D) $\cos^{-1}(\frac{1}{\sqrt{3}})$

Question 19. The planes $x+y+z=1$ and $2x+2y+2z=3$ are:

(A) Parallel

(B) Perpendicular

(C) Coincident

(D) Intersecting

Question 20. The angle between the line $\frac{x-5}{0} = \frac{y-1}{1} = \frac{z-3}{0}$ and the y-axis is:

(A) $0^\circ$

(B) $30^\circ$

(C) $45^\circ$

(D) $90^\circ$

Question 21. The condition for a line $\vec{r} = \vec{a} + \lambda \vec{b}$ to lie in the plane $\vec{r} \cdot \vec{n} = d$ is:

(A) $\vec{b} \cdot \vec{n} = 0$ and $\vec{a} \cdot \vec{n} = d$

(B) $\vec{b} \cdot \vec{n} \neq 0$ and $\vec{a} \cdot \vec{n} = d$

(C) $\vec{b} \cdot \vec{n} = 0$ and $\vec{a} \cdot \vec{n} \neq d$

(D) $\vec{b} \cdot \vec{n} \neq 0$ and $\vec{a} \cdot \vec{n} \neq d$

Question 22. If two lines are perpendicular, the dot product of their direction vectors is:

(A) 1

(B) -1

(C) 0

(D) $|\vec{b}_1||\vec{b}_2|$

Question 1. The shortest distance between two skew lines $\vec{r} = \vec{a}_1 + \lambda \vec{b}_1$ and $\vec{r} = \vec{a}_2 + \mu \vec{b}_2$ is given by:

(A) $\frac{|(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2)|}{|\vec{b}_1 \times \vec{b}_2|}$

(B) $\frac{|(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 + \vec{b}_2)|}{|\vec{b}_1 + \vec{b}_2|}$

(C) $\frac{|(\vec{a}_2 - \vec{a}_1) \times (\vec{b}_1 \cdot \vec{b}_2)|}{|\vec{b}_1 \cdot \vec{b}_2|}$

(D) $\frac{|(\vec{a}_2 - \vec{a}_1) \times (\vec{b}_1 \times \vec{b}_2)|}{|\vec{b}_1 \times \vec{b}_2|}$

Question 2. The shortest distance between two parallel lines $\vec{r} = \vec{a}_1 + \lambda \vec{b}$ and $\vec{r} = \vec{a}_2 + \mu \vec{b}$ is given by:

(A) $\frac{|(\vec{a}_2 - \vec{a}_1) \cdot \vec{b}|}{|\vec{b}|}$

(B) $\frac{|(\vec{a}_2 - \vec{a}_1) \times \vec{b}|}{|\vec{b}|}$

(C) $\frac{|(\vec{a}_2 - \vec{a}_1) \cdot \vec{b}|}{|\vec{a}_2 - \vec{a}_1|}$

(D) $\frac{|(\vec{a}_2 - \vec{a}_1) \times \vec{b}|}{|\vec{a}_2 - \vec{a}_1|}$

Question 3. The distance of a point P with position vector $\vec{p}$ from the plane $\vec{r} \cdot \vec{n} = d$ is given by:

(A) $\frac{|\vec{p} \cdot \vec{n} - d|}{|\vec{n}|}$

(B) $\frac{|\vec{p} \cdot \vec{n} + d|}{|\vec{n}|}$

(C) $\frac{|\vec{p} \times \vec{n} - d|}{|\vec{n}|}$

(D) $\frac{|\vec{p} \cdot \vec{n} - d|}{|d|}$

Question 4. The distance of a point $(x_1, y_1, z_1)$ from the plane $Ax + By + Cz + D = 0$ is given by:

(A) $\frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2+B^2+C^2}}$

(B) $\frac{Ax_1 + By_1 + Cz_1 + D}{\sqrt{A^2+B^2+C^2}}$

(C) $\frac{|Ax_1 + By_1 + Cz_1|}{\sqrt{A^2+B^2+C^2}}$

(D) $\frac{|D|}{\sqrt{A^2+B^2+C^2}}$

Question 5. The shortest distance between two skew lines is the length of the line segment which is perpendicular to:

(A) Both lines.

(B) Only the first line.

(C) Only the second line.

(D) The line joining the starting points.

Question 6. The distance of the point $(1, -2, 3)$ from the plane $x - y + z = 5$ is:

(A) 0

(B) $\frac{|1 - (-2) + 3 - 5|}{\sqrt{1^2+(-1)^2+1^2}} = \frac{|1+2+3-5|}{\sqrt{3}} = \frac{1}{\sqrt{3}}$

(C) $\frac{1}{\sqrt{3}}$

(D) $\frac{2}{\sqrt{3}}$

Question 7. The shortest distance between the line $\frac{x-1}{1} = \frac{y-2}{-1} = \frac{z-3}{1}$ and the line $\frac{x-2}{2} = \frac{y-3}{-1} = \frac{z-4}{1}$ is:

(A) 0

(B) $\frac{1}{\sqrt{3}}$

(C) $\frac{1}{\sqrt{6}}$

(D) $\frac{2}{\sqrt{3}}$

Question 8. The shortest distance between the parallel lines $\vec{r} = \hat{i} + 2\hat{j} - 4\hat{k} + \lambda (2\hat{i} + 3\hat{j} + 6\hat{k})$ and $\vec{r} = 3\hat{i} + 3\hat{j} - 5\hat{k} + \mu (2\hat{i} + 3\hat{j} + 6\hat{k})$ is:

(A) $\frac{\sqrt{19}}{7}$

(B) $\frac{\sqrt{17}}{7}$

(C) $\frac{\sqrt{23}}{7}$

(D) $\frac{\sqrt{26}}{7}$

Question 9. The distance between the planes $2x - y + 2z = 3$ and $2x - y + 2z = 9$ is:

(A) 6

(B) 3

(C) 2

(D) $\frac{|9-3|}{\sqrt{2^2+(-1)^2+2^2}} = \frac{6}{3} = 2$

Question 10. The distance of the point $(0,0,0)$ from the plane $3x - 4y + 12z + 13 = 0$ is:

(A) 1

(B) $\frac{13}{\sqrt{3^2+(-4)^2+12^2}} = \frac{13}{\sqrt{9+16+144}} = \frac{13}{\sqrt{169}} = \frac{13}{13} = 1$

(C) 13

(D) $\frac{13}{\sqrt{169}}$

Question 11. Skew lines are lines that are:

(A) Parallel and distinct.

(B) Intersecting at a point.

(C) Neither parallel nor intersecting.

(D) Coincident.

Question 12. The shortest distance between two parallel planes $Ax + By + Cz + D_1 = 0$ and $Ax + By + Cz + D_2 = 0$ is:

(A) $\frac{|D_2 - D_1|}{\sqrt{A^2+B^2+C^2}}$

(B) $\frac{|D_1 - D_2|}{A^2+B^2+C^2}$

(C) $|D_1 - D_2|$

(D) 0

Question 13. The distance of the point $(1, 2, -1)$ from the plane $x - 2y + 4z + 5 = 0$ is:

(A) 0

(B) $\frac{|1 - 2(2) + 4(-1) + 5|}{\sqrt{1^2+(-2)^2+4^2}} = \frac{|1 - 4 - 4 + 5|}{\sqrt{1+4+16}} = \frac{|-2|}{\sqrt{21}} = \frac{2}{\sqrt{21}}$

(C) $\frac{2}{\sqrt{21}}$

(D) $\frac{2}{21}$

Question 14. The shortest distance between the z-axis and the line $\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ is:

(A) $\frac{1}{\sqrt{5}}$

(B) $\frac{2}{\sqrt{5}}$

(C) $\frac{3}{\sqrt{5}}$

(D) $\frac{4}{\sqrt{5}}$

Question 15. If two lines are intersecting, their shortest distance is:

(A) $|\vec{a}_2 - \vec{a}_1|$

(B) $|\vec{b}_1 \times \vec{b}_2|$

(C) 1

(D) 0

Question 16. The distance of the point $(1, 0, 0)$ from the plane $2x + y + z = 3$ is:

(A) 0

(B) $\frac{|2(1) + 0 + 0 - 3|}{\sqrt{2^2+1^2+1^2}} = \frac{|-1|}{\sqrt{6}} = \frac{1}{\sqrt{6}}$

(C) $\frac{1}{\sqrt{6}}$

(D) $\frac{2}{\sqrt{6}}$

Question 17. The shortest distance between the line $\vec{r} = \hat{i} + \lambda \hat{k}$ and the x-axis ($\vec{r} = \mu \hat{i}$) is:

(A) 0

(B) 1

(C) $|\hat{i}| = 1$

(D) $\sqrt{1^2+1^2} = \sqrt{2}$

Question 18. The distance of a point P from a line L is the length of the perpendicular segment from P to L. This perpendicular segment meets the line at a point Q. The distance is $|\vec{PQ}|$. If the line passes through A with direction $\vec{b}$, and P has position vector $\vec{p}$, the distance formula involves:

(A) Cross product $|\vec{AP} \times \vec{b}|/|\vec{b}|$.

(B) Dot product $|\vec{AP} \cdot \vec{b}|/|\vec{b}|$.

(C) Scalar triple product.

(D) Distance formula between two points.

Question 19. If the shortest distance between two lines is zero, the lines are either intersecting or:

(A) Skew

(B) Parallel

(C) Perpendicular

(D) On the same plane

Question 20. The distance of the plane $x=5$ from the origin is:

(A) 0

(B) 5

(C) $\sqrt{5}$

(D) $\frac{1}{5}$