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:

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

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

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

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

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