represent a vector as two perpendicular components

Scalars and \cdot ectors

Learning Objective

By the end of this lesson you should be able to represent any vector as two perpendicular components.

Key Definitions

• Scalar: a quantity that has magnitude only (e.g., mass, temperature, time).
• Vector: a quantity that has both magnitude and direction (e.g., displacement, velocity, force).

Properties of \cdot ectors

1. Direction is indicated by an arrow or a unit vector.
2. Magnitude is always a non‑negative number.
3. Vectors can be added, subtracted, and multiplied by scalars.

Representing a \cdot ector in Two Dimensions

Consider a vector $\vec{A}$ lying in the $xy$‑plane. It can be expressed as the sum of two perpendicular components along the $x$‑ and $y$‑axes:

$$\vec{A}=A_x\hat{i}+A_y\hat{j}$$

where:

• $A_x$ is the component of $\vec{A}$ along the $x$‑axis.
• $A_y$ is the component of $\vec{A}$ along the $y$‑axis.
• $\hat{i}$ and $\hat{j}$ are unit vectors in the $x$‑ and $y$‑directions respectively.

Finding the Components

If the magnitude $A$ and the angle $\theta$ (measured from the positive $x$‑axis) are known, the components are obtained using trigonometry:

$$A_x = A\cos\theta$$ $$A_y = A\sin\theta$$

The reverse relations give the magnitude and direction from the components:

$$A = \sqrt{A_x^{2}+A_y^{2}}$$ $$\theta = \tan^{-1}\!\left(\frac{A_y}{A_x}\right)$$

Example

Given a force $\vec{F}$ of magnitude $50\ \text{N}$ acting $30^{\circ}$ above the positive $x$‑axis, find its components.

1. Calculate $F_x = 50\cos30^{\circ}=43.3\ \text{N}$.
2. Calculate $F_y = 50\sin30^{\circ}=25.0\ \text{N}$.
3. Thus $\vec{F}=43.3\hat{i}+25.0\hat{j}\ \text{N}$.

Common \cdot ectors in Physics

Vector Quantity	Symbol	Typical Units	Typical Components
Displacement	$\vec{s}$	metre (m)	$s_x$, $s_y$
Velocity	$\vec{v}$	metre per second (m s⁻¹)	$v_x$, $v_y$
Acceleration	$\vec{a}$	metre per second squared (m s⁻²)	$a_x$, $a_y$
Force	$\vec{F}$	newton (N)	$F_x$, $F_y$

Summary

Any vector in a plane can be uniquely described by two perpendicular components. This representation simplifies vector addition, subtraction, and the application of Newton’s laws in two dimensions.