Angle Between Vectors
What is the Angle Between Two Vectors?
When two vectors originate from the same point, they form an angle between them. This angle, denoted by $\theta$, tells us how "aligned" or "opposed" the two vectors are.
Given two vectors $\vec{a}$ and $\vec{b}$, the angle between them is the smallest angle formed when both vectors are placed tail-to-tail (starting from the same point). This angle always satisfies $0° \leq \theta \leq 180°$.
The Dot Product Formula
The key tool for finding the angle between vectors is the dot product (also called the scalar product).
Definition: Dot Product
For two vectors $\vec{a} = (a_1, a_2)$ and $\vec{b} = (b_1, b_2)$:
$$\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2$$Definition: Magnitude of a Vector
The magnitude (length) of vector $\vec{a} = (a_1, a_2)$ is:
$$|\vec{a}| = \sqrt{a_1^2 + a_2^2}$$⭐ The Angle Formula
The angle $\theta$ between vectors $\vec{a}$ and $\vec{b}$ is given by:
$$\cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| \cdot |\vec{b}|}$$Therefore:
$$\theta = \cos^{-1}\left(\frac{a_1 b_1 + a_2 b_2}{\sqrt{a_1^2 + a_2^2} \cdot \sqrt{b_1^2 + b_2^2}}\right)$$Special Cases to Remember
| Condition | Angle $\theta$ | Meaning |
|---|---|---|
| $\vec{a} \cdot \vec{b} > 0$ | $0° < \theta < 90°$ | Vectors point in a similar direction (acute) |
| $\vec{a} \cdot \vec{b} = 0$ | $\theta = 90°$ | Vectors are perpendicular (orthogonal) |
| $\vec{a} \cdot \vec{b} < 0$ | $90° < \theta < 180°$ | Vectors point in opposite-ish directions (obtuse) |
| $\cos\theta = 1$ | $\theta = 0°$ | Vectors are parallel (same direction) |
| $\cos\theta = -1$ | $\theta = 180°$ | Vectors are anti-parallel (opposite direction) |