Vector Algebra: Introduction and Basic Operations

Introduction to Vectors (Definition, Magnitude, Direction)

In physics and mathematics, various physical quantities are encountered. These quantities can be broadly classified into two fundamental categories based on whether they possess direction along with magnitude:

• Scalars: Physical quantities that are completely described by their magnitude alone. Magnitude refers to the size or numerical value of the quantity, along with appropriate units. Scalars do not have a direction associated with them. Examples of scalar quantities include:
  • Distance
  • Speed
  • Mass
  • Temperature
  • Time
  • Density
  • Volume
  • Energy
  • Work
  Scalar quantities are added, subtracted, multiplied, and divided according to the rules of ordinary arithmetic.
• Vectors: Physical quantities that possess both magnitude and direction. To fully describe a vector quantity, both its numerical value (magnitude) and its orientation or direction in space are required. Examples of vector quantities include:
  • Displacement
  • Velocity
  • Acceleration
  • Force
  • Momentum
  • Electric Field
  • Gravitational Field Strength
  • Torque
  Vector quantities obey specific rules of vector algebra for operations like addition, subtraction, and multiplication, which take into account both magnitude and direction.

Definition of a Vector

A vector is a mathematical or physical quantity characterized by both a magnitude and a direction. Geometrically, a vector is often visualized as a directed line segment in space.

Representation of a Vector

Vectors can be represented in two primary ways:

Geometric Representation: Directed Line Segment

Geometrically, a vector is represented by a directed line segment, which is a line segment with an arrowhead at one end.

• The starting point of the directed line segment is called the initial point or tail of the vector.
• The endpoint with the arrowhead is called the terminal point or head of the vector.
• The length of the directed line segment represents the magnitude of the vector. A longer segment indicates a greater magnitude.
• The arrowhead indicates the direction of the vector, pointing from the initial point towards the terminal point.

If a vector is represented by a directed line segment from an initial point A to a terminal point B, it is denoted as $\vec{AB}$. The initial point is A and the terminal point is B.

Symbolic Representation

In mathematical notation, vectors are typically denoted by:

• Lowercase letters with an arrow above them, such as $\vec{a}, \vec{b}, \vec{v}, \vec{F}$. This notation explicitly shows the vector nature.
• Boldface lowercase letters, such as $\mathbf{a}, \mathbf{b}, \mathbf{v}, \mathbf{F}$. This is common in printed text.
• Sometimes, capital letters with an arrow are used for vectors representing displacement between points, like $\vec{PQ}$.

Position Vector

The position vector of a point P relative to a fixed origin O in space is the vector $\vec{OP}$. It uniquely specifies the position of point P with respect to the origin. The tail of the position vector is at the origin O, and its head is at the point P.

If the coordinates of the origin are $O = (0, 0, 0)$ and the coordinates of the point P are $(x, y, z)$ in a 3D Cartesian coordinate system, then the position vector of P, $\vec{OP}$, can be represented in terms of its components (which will be discussed in detail in a later section) as:

$\vec{OP} = x\hat{i} + y\hat{j} + z\hat{k}$

where $\hat{i}, \hat{j}, \hat{k}$ are the standard unit vectors along the positive x, y, and z axes, respectively.

Magnitude of a Vector

The magnitude (also called the length, norm, or modulus) of a vector $\vec{a}$ is a non-negative scalar quantity representing its "size" or how long the directed line segment is. It tells us "how much" of the quantity the vector represents.

• The magnitude of a vector $\vec{a}$ is denoted by $|\vec{a}|$. If the vector is $\vec{AB}$, its magnitude is denoted by $|\vec{AB}|$. Sometimes, especially in physics, the magnitude of a vector $\vec{a}$ is simply denoted by the corresponding non-bold or non-arrowed letter, $a$.
• The magnitude is always non-negative: $|\vec{a}| \ge 0$.
• The only vector with zero magnitude is the zero vector.
• If a vector $\vec{a}$ is given in component form in a 3D Cartesian system as $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$, its magnitude is calculated using the 3D distance formula from the origin $(0,0,0)$ to the point $(a_1, a_2, a_3)$ assuming the vector starts at the origin (which is often the case when thinking about components):

  $|\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}$

• If the vector is defined by two points, say $\vec{AB}$ where $A = (x_1, y_1, z_1)$ and $B = (x_2, y_2, z_2)$, the components of $\vec{AB}$ are $(x_2-x_1, y_2-y_1, z_2-z_1)$. Therefore, the magnitude is:

  $|\vec{AB}| = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$

Direction of a Vector

The direction of a vector specifies its orientation in space. It tells us "which way" the quantity acts or points. The direction can be described in several ways:

• Geometrically: By the orientation of the directed line segment and its arrowhead.
• By Angles: In a coordinate system, the direction of a vector $\vec{a}$ is often described by the angles it makes with the positive x, y, and z axes. Let these angles be $\alpha, \beta, \gamma$ respectively. These are called the direction angles. The cosines of these angles, $\cos \alpha, \cos \beta, \cos \gamma$, are called the direction cosines of the vector. They are commonly denoted by $l, m, n$. There is a fundamental relationship between direction cosines: $l^2 + m^2 + n^2 = 1$. Direction cosines provide a unique way to specify the direction of a vector.
• By a Unit Vector: As we will see, a unit vector pointing in the same direction as $\vec{a}$ serves as a pure indicator of the direction of $\vec{a}$.

An important concept in vector algebra is that vectors are often considered free vectors, meaning they can be translated (moved parallel to themselves) anywhere in space without changing their identity, as long as their magnitude and direction remain the same. The exception is the position vector, which is fixed relative to the origin it's defined from.

Types of Vectors (Zero, Unit, Coinitial, Collinear, Equal)

Vectors can be classified into different types based on their properties such as magnitude, direction, and their relative position or orientation with respect to other vectors.

Zero Vector (or Null Vector)

• Definition: A vector whose magnitude is zero is called the zero vector or null vector.
• Notation: It is denoted by $\vec{0}$ or $\mathbf{0}$.
• Properties:
  • The initial and terminal points of a zero vector coincide. For example, $\vec{AA}$ is a zero vector.
  • Its magnitude is zero: $|\vec{0}| = 0$.
  • The direction of the zero vector is undefined or indeterminate. It can be considered to have any direction.
  • Adding the zero vector to any vector $\vec{a}$ results in $\vec{a}$: $\vec{a} + \vec{0} = \vec{a}$.
  • Multiplying the zero vector by any scalar $k$ results in the zero vector: $k\vec{0} = \vec{0}$.
  • Multiplying any vector $\vec{a}$ by the scalar 0 results in the zero vector: $0 \vec{a} = \vec{0}$.
• Example: The velocity vector of a stationary object is a zero vector.

Unit Vector

• Definition: A vector whose magnitude is exactly one unit is called a unit vector.
• Purpose: Unit vectors are primarily used to specify or indicate a direction in space. They carry directional information without any magnitude component other than 1.
• Notation: A unit vector in the direction of a given non-zero vector $\vec{a}$ is typically denoted by $\hat{a}$ (read as "a cap" or "a hat").
• Formula: For any non-zero vector $\vec{a}$, the unit vector $\hat{a}$ in the same direction is obtained by dividing the vector $\vec{a}$ by its magnitude $|\vec{a}|$.

  $\hat{a} = \frac{\vec{a}}{|\vec{a}|}$

  This formula essentially "normalizes" the vector to unit length while preserving its direction.
• From the above formula, we can express any non-zero vector as the product of its magnitude and its unit vector:

  $\vec{a} = |\vec{a}| \hat{a}$

  This shows that any vector is a scalar multiple of its corresponding unit vector, where the scalar is the magnitude.
• Standard Unit Vectors: In a 3D Cartesian coordinate system, there are three standard unit vectors that point along the positive directions of the x, y, and z axes. These are fundamental for representing vectors in component form:
  • $\hat{i}$ is the unit vector along the positive x-axis. In component form, $\hat{i} = (1, 0, 0)$. Its magnitude is $|\hat{i}| = \sqrt{1^2 + 0^2 + 0^2} = 1$.
  • $\hat{j}$ is the unit vector along the positive y-axis. In component form, $\hat{j} = (0, 1, 0)$. Its magnitude is $|\hat{j}| = \sqrt{0^2 + 1^2 + 0^2} = 1$.
  • $\hat{k}$ is the unit vector along the positive z-axis. In component form, $\hat{k} = (0, 0, 1)$. Its magnitude is $|\hat{k}| = \sqrt{0^2 + 0^2 + 1^2} = 1$.
  Any vector $\vec{a}$ with components $(a_1, a_2, a_3)$ can be uniquely expressed as a linear combination of these standard unit vectors: $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$.

Coinitial Vectors

• Definition: Two or more vectors are called coinitial vectors if they all have the same initial point (the same starting point or tail).
• Geometric Representation: Graphically, coinitial vectors appear as arrows originating from the same point.
• For example, if vectors $\vec{a}, \vec{b}, \vec{c}$ all start from point P, they are coinitial vectors originating from P.

Collinear Vectors

• Definition: Two or more vectors are said to be collinear if they are parallel to the same line, regardless of their magnitudes or directions (whether they point in the same or opposite sense).
• Geometric Representation: Collinear vectors lie on the same straight line or on parallel straight lines.
• Condition for Collinearity: Two non-zero vectors $\vec{a}$ and $\vec{b}$ are collinear if and only if one vector can be expressed as a scalar multiple of the other. That is, there exists a non-zero scalar $\lambda$ such that:

  $\vec{a} = \lambda \vec{b}$

  • If $\lambda > 0$, the vectors $\vec{a}$ and $\vec{b}$ are parallel and point in the same direction.
  • If $\lambda < 0$, the vectors $\vec{a}$ and $\vec{b}$ are anti-parallel and point in opposite directions.
• The zero vector $\vec{0}$ is considered collinear with any vector $\vec{a}$, since $\vec{0} = 0 \cdot \vec{a}$.
• If vectors are given in component form, say $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ and $\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$ (with $\vec{b} \ne \vec{0}$), then $\vec{a}$ and $\vec{b}$ are collinear if and only if their corresponding components are proportional:

  $\frac{a_1}{b_1} = \frac{a_2}{b_2} = \frac{a_3}{b_3} = \lambda$ (assuming no component of $\vec{b}$ is zero)

  If a component of $\vec{b}$ is zero, the corresponding component of $\vec{a}$ must also be zero for them to be collinear.

Equal Vectors

• Definition: Two vectors $\vec{a}$ and $\vec{b}$ are said to be equal if and only if they have the same magnitude AND the same direction.
• Notation: $\vec{a} = \vec{b}$.
• Geometric Representation: Equal vectors are represented by parallel directed line segments of the same length pointing in the same sense. They do not necessarily have the same initial or terminal points (unless they are position vectors from the same origin).
• Condition in Component Form: 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 $\vec{a} = \vec{b}$ if and only if their corresponding components are equal:

  $a_1 = b_1$

  $a_2 = b_2$

  $a_3 = b_3$

• Equal vectors represent the same physical effect. For instance, two equal force vectors applied to the same object will have the same effect on its motion (if applied at the same point or if the object is a rigid body).

Negative of a Vector

• Definition: The negative of a vector $\vec{a}$, denoted by $-\vec{a}$, is a vector that has the same magnitude as $\vec{a}$ but points in the opposite direction.
• Properties:
  • Magnitude: $|-\vec{a}| = |\vec{a}|$.
  • Direction: $-\vec{a}$ is anti-parallel to $\vec{a}$.
  • Addition: Adding a vector to its negative results in the zero vector: $\vec{a} + (-\vec{a}) = \vec{0}$. This is the basis for vector subtraction.
• Geometric Representation: If $\vec{a}$ is represented by the directed line segment $\vec{AB}$ (from A to B), then $-\vec{a}$ is represented by the directed line segment $\vec{BA}$ (from B to A).
• In Component Form: If $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$, then its negative is obtained by negating each component:

  $-\vec{a} = (-a_1)\hat{i} + (-a_2)\hat{j} + (-a_3)\hat{k}$

Vector Algebra (Addition, Subtraction, Multiplication by a Scalar)

Just like numbers, vectors can be combined using algebraic operations. The fundamental operations in vector algebra are vector addition, vector subtraction, and the multiplication of a vector by a scalar. These operations allow us to manipulate and combine vectors to solve problems in geometry and physics.

Vector Addition

The sum of two or more vectors results in a single vector, known as the resultant vector. Vector addition is not the same as scalar addition because the directions of the vectors must be taken into account. There are two primary geometric laws for adding vectors:

Triangle Law of Vector Addition

If two vectors $\vec{a}$ and $\vec{b}$ are represented in magnitude and direction by two sides of a triangle taken in order (meaning the initial point of the second vector coincides with the terminal point of the first vector), then their sum or resultant vector, $\vec{a} + \vec{b}$, is represented in magnitude and direction by the third side of the triangle taken in the opposite order (from the initial point of the first vector to the terminal point of the second vector).

Let $\vec{AB} = \vec{a}$ be a vector from point A to point B, and $\vec{BC} = \vec{b}$ be a vector from point B to point C. According to the Triangle Law, the vector $\vec{AC}$ represents the sum $\vec{a} + \vec{b}$.

Parallelogram Law of Vector Addition

If two vectors $\vec{a}$ and $\vec{b}$ are represented in magnitude and direction by the two adjacent sides of a parallelogram drawn from a common initial point, then their sum or resultant vector, $\vec{a} + \vec{b}$, is represented in magnitude and direction by the diagonal of the parallelogram that passes through that common initial point.

Let $\vec{OA} = \vec{a}$ and $\vec{OB} = \vec{b}$ be two vectors originating from the same point O. Construct a parallelogram OACB using $\vec{OA}$ and $\vec{OB}$ as adjacent sides. According to the Parallelogram Law, the diagonal $\vec{OC}$ represents the sum $\vec{a} + \vec{b}$.

The Parallelogram Law is consistent with the Triangle Law. In the parallelogram OACB, $\vec{OA} = \vec{a}$ and $\vec{AC} = \vec{OB} = \vec{b}$ (since opposite sides of a parallelogram are equal and parallel). Applying the Triangle Law to triangle OAC, we get $\vec{OC} = \vec{OA} + \vec{AC} = \vec{a} + \vec{b}$.

Addition in Component Form

When vectors are expressed in terms of their components using standard unit vectors $\hat{i}, \hat{j}, \hat{k}$, vector addition becomes straightforward. If we have two vectors:

Let $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$

Let $\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$

Then their sum $\vec{a} + \vec{b}$ is found by adding the corresponding components:

$$ \vec{a} + \vec{b} = (a_1 + b_1)\hat{i} + (a_2 + b_2)\hat{j} + (a_3 + b_3)\hat{k} $$

This is the most convenient way to add vectors computationally.

Vector Subtraction

The subtraction of a vector $\vec{b}$ from a vector $\vec{a}$ is defined as the addition of vector $\vec{a}$ and the negative (or additive inverse) of vector $\vec{b}$.

Recall that $-\vec{b}$ is a vector with the same magnitude as $\vec{b}$ but pointing in the opposite direction.

Geometrically, using the Parallelogram Law (vectors $\vec{OA} = \vec{a}$ and $\vec{OB} = \vec{b}$ originating from O), the vector $\vec{BA}$ represents the difference $\vec{a} - \vec{b}$. This is because $\vec{OB} + \vec{BA} = \vec{OA}$ by the Triangle Law, which implies $\vec{BA} = \vec{OA} - \vec{OB} = \vec{a} - \vec{b}$. Note that $\vec{OC}$ is $\vec{a} + \vec{b}$ and $\vec{BA}$ is $\vec{a} - \vec{b}$. These are the two diagonals of the parallelogram formed by $\vec{a}$ and $\vec{b}$ as adjacent sides from a common origin.

Subtraction in Component Form

Similar to addition, subtraction is performed component-wise using the definition $\vec{a} - \vec{b} = \vec{a} + (-\vec{b})$. 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 $-\vec{b} = (-b_1)\hat{i} + (-b_2)\hat{j} + (-b_3)\hat{k}$.

So, their difference is:

$$ \vec{a} - \vec{b} = (a_1 - b_1)\hat{i} + (a_2 - b_2)\hat{j} + (a_3 - b_3)\hat{k} $$

Multiplication of a Vector by a Scalar

Multiplying a vector $\vec{a}$ by a scalar (a real number) $k$ results in a new vector, denoted by $k\vec{a}$. This operation scales the magnitude of the vector and may change its direction depending on the sign of the scalar.

• Magnitude: The magnitude of the resulting vector $k\vec{a}$ is the absolute value of the scalar $k$ multiplied by the magnitude of the original vector $\vec{a}$.

  $|k\vec{a}| = |k| |\vec{a}|$

• Direction:
  • If $k > 0$ (positive scalar), the vector $k\vec{a}$ has the same direction as $\vec{a}$. For example, $2\vec{a}$ points in the same direction as $\vec{a}$ but is twice as long.
  • If $k < 0$ (negative scalar), the vector $k\vec{a}$ has the opposite direction to $\vec{a}$. For example, $-\vec{a}$ points in the opposite direction with the same length, and $-3\vec{a}$ points in the opposite direction and is three times as long.
  • If $k = 0$, then $k\vec{a} = 0 \cdot \vec{a} = \vec{0}$ (the zero vector). The zero vector has zero magnitude and an undefined direction.

Geometrically, scalar multiplication stretches or shrinks the vector along its original line of action and may flip its orientation if the scalar is negative. The resulting vector $k\vec{a}$ is always collinear with the original vector $\vec{a}$.

Scalar Multiplication in Component Form

When a vector is in component form, scalar multiplication is performed by multiplying each component of the vector by the scalar $k$. If $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ and $k$ is a scalar, then:

$$ k\vec{a} = k(a_1\hat{i} + a_2\hat{j} + a_3\hat{k}) = (k a_1)\hat{i} + (k a_2)\hat{j} + (k a_3)\hat{k} $$

This is a straightforward element-wise multiplication.

Answer:

Properties of Vector Addition and Scalar Multiplication

Vector addition and scalar multiplication obey several algebraic properties that are similar to those of real numbers. These properties are fundamental to manipulating vector equations and understanding vector spaces.

Let $\vec{a}, \vec{b}, \vec{c}$ be any three vectors, and let $k, m$ be any two scalars (real numbers).

Properties of Vector Addition

1. Commutativity of Vector Addition:

   The order in which two vectors are added does not affect the result.

   $\vec{a} + \vec{b} = \vec{b} + \vec{a}$

   This property can be easily visualized using the Parallelogram Law, where both $\vec{a} + \vec{b}$ and $\vec{b} + \vec{a}$ correspond to the same diagonal of the parallelogram.

2. Associativity of Vector Addition:

   When adding three or more vectors, the grouping of the vectors does not affect the sum. This allows us to add multiple vectors sequentially without using parentheses.

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

   This property implies that we can simply write $\vec{a} + \vec{b} + \vec{c}$ without any ambiguity about the order of operations. Geometrically, this can be shown using the polygon law of vector addition.

3. Existence of Additive Identity:

   There exists a unique vector, the zero vector $\vec{0}$, such that when it is added to any vector $\vec{a}$, the vector $\vec{a}$ remains unchanged.

   $\vec{a} + \vec{0} = \vec{0} + \vec{a} = \vec{a}$

   The zero vector acts as the additive identity element in vector addition.

4. Existence of Additive Inverse:

   For every vector $\vec{a}$, there exists a unique vector, the negative of $\vec{a}$ (denoted by $-\vec{a}$), such that when added to $\vec{a}$, the result is the zero vector.

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

   The vector $-\vec{a}$ acts as the additive inverse of $\vec{a}$. Vector subtraction $\vec{a} - \vec{b}$ is defined using this property as $\vec{a} + (-\vec{b})$.

These four properties establish that the set of all vectors (in a given space, like 2D or 3D) forms an Abelian group under the operation of vector addition.

Properties of Scalar Multiplication

Scalar multiplication interacts with vector addition in the following ways:

1. Distributivity over Vector Addition:

   A scalar can be distributed over the sum of two vectors.

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

2. Distributivity over Scalar Addition:

   A vector can be distributed over the sum of two scalars.

   $(k + m)\vec{a} = k\vec{a} + m\vec{a}$

3. Associativity of Scalar Multiplication:

   The multiplication of scalars is associative when one scalar is multiplied by the result of another scalar multiplying a vector.

   $k(m\vec{a}) = (km)\vec{a}$

   This means multiplying a vector by a product of scalars is the same as multiplying it sequentially by each scalar.

4. Multiplication by the Multiplicative Identity Scalar:

   Multiplying any vector by the scalar 1 results in the original vector.

   $1 \cdot \vec{a} = \vec{a}$

   The scalar 1 acts as the multiplicative identity for scalar multiplication of vectors.

5. Multiplication by the Zero Scalar:

   Multiplying any vector by the scalar 0 results in the zero vector.

   $0 \cdot \vec{a} = \vec{0}$

These properties, combined with the properties of vector addition, are the defining axioms for a mathematical structure called a vector space. In this context, vectors are the elements of the space, and scalars are elements of the field (in this case, the field of real numbers $\mathbb{R}$). This structure is fundamental in linear algebra and many areas of physics and engineering.