How Can You Find the Angle Between Two Vectors?

Two vectors making up an angle

The angle between two vectors is always an angle $α \in [0^{\circ}, 180^{\circ}]$ .

Formula

\cos α = \frac{\vec{u} \cdot \vec{v}}{| \vec{u} | \cdot | \vec{v} |}, α \in [0^{\circ}, 180^{\circ}]

Example 1

Find the angle between the vectors $(4, 1)$ and $(2, - 3)$ .

You insert the numbers into the formula, which gives you

\begin{array}{l} \cos α & = \frac{(4, 1) \cdot (2, - 3)}{| (4, 1) | \cdot | (2, - 3) |} \\ = \frac{8 - 3}{\sqrt{4^{2} + 1^{2}} \cdot \sqrt{2^{2} + {(- 3)}^{2}}} \\ = \frac{5}{\sqrt{17} \cdot \sqrt{13}} \\ \approx \frac{5}{14.86} \\ \approx 0.336 . \end{array}

That means the angle is

α = \cos^{- 1} (0.336) \approx {70.3}^{\circ} .

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