Angles between vectors

Japanese school year: Math C

What you learn

Learn how to find the cosine of the angle between two vectors using their magnitudes and dot product. This method is widely used to analyze directional relationships in two- and three-dimensional spaces. Familiarity with vector magnitudes, dot product calculations, and fundamental trigonometry is required.

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Key points

This formula determines the angle $\theta$ between two vectors $\vec{a}$ and $\vec{b}$ . Dividing their dot product $\vec{a} \cdot \vec{b}$ by the product of their magnitudes $|\vec{a}||\vec{b}|$ gives $\cos\theta$ .

\cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}

This formula calculates $\cos\theta$ directly from the coordinates of two vectors, $(a_1, a_2)$ and $(b_1, b_2)$ . The numerator represents their dot product, while the denominator is the product of their lengths.

\cos\theta = \frac{a_1 b_1 + a_2 b_2}{\sqrt{a_1^2 + a_2^2}\sqrt{b_1^2 + b_2^2}}

This inequality defines the valid range for the angle $\theta$ between two vectors. By convention, the smaller angle between the two directions is chosen, restricting $\theta$ between $0$ and $\pi$ radians ( $0^\circ$ to $180^\circ$ ).

0 \le \theta \le \pi

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Practice 1Coming soon