Scalars, Vectors, and Basic Operations

Representing Physical Quantities (Scalar vs Vector)

Physical quantities can be broadly classified into two main categories based on how they are described: scalars and vectors.

Scalar Quantities

A scalar quantity is completely described by a single number (its magnitude) along with the appropriate unit. It does not have a direction associated with it.

• Definition: Has magnitude only.
• Examples:
  • Mass: 5 kg
  • Temperature: 25 degrees Celsius
  • Time: 10 seconds
  • Distance: 5 km (the total path length)
  • Speed: 60 km/h
  • Energy: 100 Joules
  • Density: 1000 kg/m³
• Operations: Scalars are manipulated using ordinary rules of algebra (addition, subtraction, multiplication, division).

Vector Quantities

A vector quantity requires both a magnitude and a direction to be completely described. The direction is crucial for defining the quantity's effect or behavior.

• Definition: Has both magnitude and direction.
• Representation: Vectors are typically represented by arrows. The length of the arrow represents the magnitude (scaled appropriately), and the direction of the arrow indicates the direction of the vector. Vectors are often denoted by bold letters (e.g., v), or letters with an arrow above them (e.g., $ \vec{v} $).
• Examples:
  • Displacement: 5 km North (distinguishes from just 5 km distance).
  • Velocity: 60 km/h East (specifies both speed and direction).
  • Force: 10 Newtons downwards.
  • Acceleration: 9.8 m/s² downwards.
  • Momentum: Mass × Velocity (has direction of velocity).
  • Electric Field: Has both strength and direction at a point in space.
  • Magnetic Field: Has both strength and direction.
• Operations: Vectors are manipulated using specific rules of vector algebra, which account for their direction. Simple algebraic addition does not apply.

Key Distinction: While distance is a scalar (total path covered), displacement is a vector (change in position from start to end, including direction). Similarly, speed is scalar, while velocity is vector.

Vector Algebra Basics (Graphical and Analytical Addition/Subtraction)

Vector algebra involves operations like addition, subtraction, and multiplication that respect the directional nature of vectors.

Vector Addition

Graphical Method (Triangle Law of Vector Addition):

1. To add two vectors, say A and B, draw vector A.
2. From the head (tip) of vector A, draw vector B.
3. The resultant vector, R ($ \mathbf{R} = \mathbf{A} + \mathbf{B} $), is the vector drawn from the tail (origin) of A to the head of B.

Graphical Method (Parallelogram Law of Vector Addition):

1. Draw both vectors A and B starting from the same origin.
2. Complete the parallelogram with A and B as adjacent sides.
3. The resultant vector, R ($ \mathbf{R} = \mathbf{A} + \mathbf{B} $), is the diagonal of the parallelogram starting from the common origin.

Analytical Method (Component Method):

This method is more precise and generally preferred for calculations.

1. Represent each vector in terms of its components along the coordinate axes (e.g., x, y, z). For 2D, a vector A can be written as $ \mathbf{A} = A_x \hat{i} + A_y \hat{j} $, where $ A_x $ and $ A_y $ are the components along the x and y axes, respectively, and $ \hat{i}, \hat{j} $ are unit vectors along the x and y axes.
2. To add vectors, add their corresponding components:

   If $ \mathbf{A} = A_x \hat{i} + A_y \hat{j} $ and $ \mathbf{B} = B_x \hat{i} + B_y \hat{j} $,

   Then $ \mathbf{R} = \mathbf{A} + \mathbf{B} = (A_x + B_x) \hat{i} + (A_y + B_y) \hat{j} $.

3. The magnitude of the resultant vector $ R $ is $ R = |\mathbf{R}| = \sqrt{R_x^2 + R_y^2} $, where $ R_x = A_x + B_x $ and $ R_y = A_y + B_y $.
4. The direction (angle $ \theta $) of the resultant vector can be found using $ \tan \theta = \frac{R_y}{R_x} $.

Vector Subtraction

Subtracting vector B from vector A ($ \mathbf{A} - \mathbf{B} $) is equivalent to adding the negative of vector B to vector A ($ \mathbf{A} + (-\mathbf{B}) $).

• Graphical Method: To find $ -\mathbf{B} $, reverse the direction of vector B while keeping its magnitude the same. Then apply the triangle or parallelogram law for A and $ -\mathbf{B} $.
• Analytical Method: Subtract the corresponding components:

  If $ \mathbf{A} = A_x \hat{i} + A_y \hat{j} $ and $ \mathbf{B} = B_x \hat{i} + B_y \hat{j} $,

  Then $ \mathbf{D} = \mathbf{A} - \mathbf{B} = (A_x - B_x) \hat{i} + (A_y - B_y) \hat{j} $.

  The magnitude and direction are calculated similarly to vector addition.

Multiplication of Vectors (Scalar and Vector Products)

There are two primary ways to multiply vectors:

1. Scalar Product (Dot Product)

The scalar product of two vectors A and B results in a scalar quantity. It is denoted by $ \mathbf{A} \cdot \mathbf{B} $.

• Definition: $ \mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \cos \theta $, where $ \theta $ is the angle between the two vectors.
• Interpretation: The dot product represents the component of one vector multiplied by the magnitude of the other vector. For example, $ \mathbf{A} \cdot \mathbf{B} $ can be interpreted as the magnitude of A times the component of B along A, or vice versa.
• Properties:
  • Commutative: $ \mathbf{A} \cdot \mathbf{B} = \mathbf{B} \cdot \mathbf{A} $
  • Distributive: $ \mathbf{A} \cdot (\mathbf{B} + \mathbf{C}) = \mathbf{A} \cdot \mathbf{B} + \mathbf{A} \cdot \mathbf{C} $
  • If $ \mathbf{A} \perp \mathbf{B} $ (vectors are perpendicular), then $ \theta = 90^\circ $ and $ \cos 90^\circ = 0 $, so $ \mathbf{A} \cdot \mathbf{B} = 0 $.
  • If $ \mathbf{A} \parallel \mathbf{B} $ (vectors are parallel), then $ \theta = 0^\circ $ and $ \cos 0^\circ = 1 $, so $ \mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}| $.
  • In terms of components: If $ \mathbf{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k} $ and $ \mathbf{B} = B_x \hat{i} + B_y \hat{j} + B_z \hat{k} $, then $ \mathbf{A} \cdot \mathbf{B} = A_x B_x + A_y B_y + A_z B_z $. (Recall $ \hat{i} \cdot \hat{i} = \hat{j} \cdot \hat{j} = \hat{k} \cdot \hat{k} = 1 $ and $ \hat{i} \cdot \hat{j} = \hat{j} \cdot \hat{k} = \hat{k} \cdot \hat{i} = 0 $).
• Applications: Calculating work done ($ W = \vec{F} \cdot \vec{d} $), power ($ P = \vec{F} \cdot \vec{v} $), magnetic flux ($ \Phi_B = \int \vec{B} \cdot d\vec{A} $).

2. Vector Product (Cross Product)

The vector product of two vectors A and B results in a third vector, denoted by $ \mathbf{A} \times \mathbf{B} $. The resulting vector is perpendicular to both A and B, and its direction is given by the right-hand rule.

• Definition: $ \mathbf{A} \times \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \sin \theta \, \hat{n} $, where $ \theta $ is the angle between the two vectors, and $ \hat{n} $ is a unit vector perpendicular to the plane containing A and B, determined by the right-hand rule.
• Magnitude: The magnitude $ |\mathbf{A} \times \mathbf{B}| $ represents the area of the parallelogram formed by vectors A and B.
• Properties:
  • Anti-commutative: $ \mathbf{A} \times \mathbf{B} = -(\mathbf{B} \times \mathbf{A}) $. The order matters.
  • Distributive: $ \mathbf{A} \times (\mathbf{B} + \mathbf{C}) = \mathbf{A} \times \mathbf{B} + \mathbf{A} \times \mathbf{C} $.
  • If $ \mathbf{A} \parallel \mathbf{B} $ or $ \mathbf{A} \text{ anti-parallel } \mathbf{B} $, then $ \theta = 0^\circ $ or $ \theta = 180^\circ $, so $ \sin \theta = 0 $. Thus, $ \mathbf{A} \times \mathbf{B} = \mathbf{0} $. The cross product of parallel vectors is zero.
  • In terms of components: If $ \mathbf{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k} $ and $ \mathbf{B} = B_x \hat{i} + B_y \hat{j} + B_z \hat{k} $, the cross product can be calculated using a determinant:

    $ \mathbf{A} \times \mathbf{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix} = (A_y B_z - A_z B_y) \hat{i} - (A_x B_z - A_z B_x) \hat{j} + (A_x B_y - A_y B_x) \hat{k} $

• Applications: Calculating torque ($ \vec{\tau} = \vec{r} \times \vec{F} $), angular momentum ($ \vec{L} = \vec{r} \times \vec{p} $), magnetic force on a moving charge ($ \vec{F} = q(\vec{v} \times \vec{B}) $).