Skip to main content
Math Vector algebra Angles between vectors

Angles between two vectors

Using the scalar product you can determine the angle between two vectors.

The following formula is used to calculate:

$\cos(\gamma) = \frac{\vec{a}\cdot\vec{b}}{|\vec{a}|\cdot|\vec{b}|}$

$\gamma = \cos^{-1}\left(\frac{\vec{a}\cdot\vec{b}}{|\vec{a}|\cdot|\vec{b}|}\right)$
i

Method

Example

Determine the angle between $\vec{a}=\begin{pmatrix}2\\6\\-3\end{pmatrix}$ and $\vec{b}=\begin{pmatrix}2\\3\\5\end{pmatrix}$.

  1. Calculate the scalar product

    $\vec{a}\cdot\vec{b}$ $=\begin{pmatrix}2\\6\\-3\end{pmatrix}\cdot\begin{pmatrix}2\\3\\5\end{pmatrix}$ $=2\cdot2+6\cdot3-3\cdot5$ $=7$
  2. Calculate the vector length

    $|\vec{a}|=\sqrt{2^2+6^2+(-3)^2}$ $=7$

    $|\vec{b}|=\sqrt{2^2+3^2+5^2}$ $=\sqrt{38}$
  3. Insert results in the formula

    $\cos(\gamma) = \frac{\vec{a}\cdot\vec{b}}{|\vec{a}|\cdot|\vec{b}|}$

    $\cos(\gamma) = \frac{7}{7\cdot\sqrt{38}}$ $= \frac{1}{\sqrt{38}}$

    $\gamma = \cos^{-1}\left(\frac{1}{\sqrt{38}}\right)$ $\approx80.66°$