Vector (Physics): Definition and Mathematical Representation

Introduction

In physics, a vector is a quantity that has both magnitude and direction. Vectors are essential in describing physical phenomena such as displacement, velocity, acceleration, force, and momentum.

Unlike scalars, which are described by a single value, vectors are represented by arrows whose length corresponds to magnitude and whose orientation indicates direction.

Definition

A vector $\vec{A}$ in component form is written as:

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

Where:

$A_{x}, A_{y}, A_{z}$ are the components of the vector along the x, y, and z axes respectively.
$\hat{i}, \hat{j}, \hat{k}$ are the unit vectors in the x, y, and z directions.

Magnitude of a Vector

The magnitude (length) of a 3D vector is given by:

$| \vec{A} | = \sqrt{A_{x}^{2} + A_{y}^{2} + A_{z}^{2}}$

For a 2D vector:

$| \vec{A} | = \sqrt{A_{x}^{2} + A_{y}^{2}}$

Direction of a Vector

The direction (angle $θ$ ) in 2D from the x-axis is:

$θ = \tan^{- 1} (\frac{A_{y}}{A_{x}})$

Vector Operations

1. Addition

$\vec{C} = \vec{A} + \vec{B} = (A_{x} + B_{x}) \hat{i} + (A_{y} + B_{y}) \hat{j}$

Graphically represented using the **head-to-tail** method or **parallelogram rule**.

2. Subtraction

$\vec{C} = \vec{A} - \vec{B} = (A_{x} - B_{x}) \hat{i} + (A_{y} - B_{y}) \hat{j}$

Equivalent to adding the negative of a vector.

3. Scalar Multiplication

$k \vec{A} = (k A_{x}) \hat{i} + (k A_{y}) \hat{j}$

Where $k$ is a scalar. If $k < 0$ , the vector direction is reversed.

4. Dot Product (Scalar Product)

$\vec{A} \cdot \vec{B} = | \vec{A} | | \vec{B} | \cos θ = A_{x} B_{x} + A_{y} B_{y} + A_{z} B_{z}$

Result is a scalar.

5. Cross Product (Vector Product)

$\vec{A} \times \vec{B} = | \vec{A} | | \vec{B} | \sin θ \hat{n}$

Where $\hat{n}$ is a unit vector perpendicular to the plane of $\vec{A}$ and $\vec{B}$ . Result is a vector.

Unit Vectors

Unit vectors have magnitude 1 and indicate direction only. Common unit vectors are:

$\hat{i}$ along x-axis
$\hat{j}$ along y-axis
$\hat{k}$ along z-axis

Example: If $\vec{v} = 3 \hat{i} + 4 \hat{j}$ , then:

$| \vec{v} | = \sqrt{3^{2} + 4^{2}} = 5$

Applications in Physics

Vectors are used to represent:

Graphical Representation

Vectors are shown as arrows:

Length = magnitude
Angle = direction
Arrows can be added/subtracted graphically

Vector (physics)

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