Direction of a Vector

1. Definition

The direction of a vector describes the orientation in space in which the vector points.

A vector is fully defined only when both are specified:

• magnitude (how large)
• direction (which way)

2. Direction in Two Dimensions

For a vector written in component form:

\(\vec{A}=(A_x , A_y , A_z)\)

the direction is typically given by the angle $\theta$ measured from the positive x-axis.

$$\tan\theta = \frac{A_y}{A_x}$$

$$\theta = \tan^{-1}\frac{A_y}{A_x}$$

3. Quadrant Consideration

The angle from tan⁡−1\tan^{-1}tan−1 must be adjusted based on the signs of the components:

• \(A_x > 0 , A_y > 0 \) → Quadrant I
• \(A_x < 0 , A_y > 0 \) → Quadrant II
• \(A_x < 0 , A_y < 0 \) → Quadrant III
• \(A_x > 0 , A_y < 0 \) → Quadrant IV

This ensures the direction is physically correct.

4. Direction Using Unit Vectors

A vector’s direction can also be expressed using a unit vector:

\(\hat{A} = \frac{\vec{A}}{|\vec{A}|}​\)

This gives a vector of magnitude 1 that points in the same direction as the original vector.

5. Direction in Three Dimensions

For a vector:

\(\vec{A}=(A_x , A_y , A_z)\)

its direction is described by the unit vector:

\(\hat{A} = \frac{A_x}{|\vec{A}|}\hat{i}+\frac{A_y}{|\vec{A}|}\hat{j}+\frac{A_z}{|\vec{A}|}\hat{k}

These ratios are called direction cosines.

6. Physical Meaning in Mechanics

Direction determines how vectors affect motion and forces.

Examples:

• velocity direction → direction of motion
• force direction → direction of acceleration
• acceleration direction → how velocity changes

Even if magnitudes are the same, different directions produce different physical outcomes.

Summary

The direction of a vector specifies its orientation in space.

In 2D:

$$\theta = \tan^{-1}\frac{A_y}{A_x}$$

It can also be expressed using a unit vector:

\(\hat{A} = \frac{\vec{A}}{|\vec{A}|}​\)​

Direction, together with magnitude, completely defines a vector in mechanics.