# Vinkelen mellom to vektorer

Vinkelen mellom to vektorer er alltid en vinkel $\alpha \in \phantom{\rule{-0.17em}{0ex}}\left[0\text{°},180\text{°}\right]$.

Formel

### Vinkelenmellomtovektorer

 $\mathrm{cos}\alpha =\frac{\stackrel{\to }{u}\cdot \stackrel{\to }{v}}{\phantom{\rule{-0.17em}{0ex}}|\stackrel{\to }{u}|\cdot \phantom{\rule{-0.17em}{0ex}}|\stackrel{\to }{v}|},\phantom{\rule{2em}{0ex}}\alpha \in \phantom{\rule{-0.17em}{0ex}}\left[0\text{°},180\text{°}\right]$

Eksempel 1

Finn vinkelen mellom vektorene $\phantom{\rule{-0.17em}{0ex}}\left[4,1\right]$ og $\phantom{\rule{-0.17em}{0ex}}\left[2,-3\right]$

Du setter inn i formelen: $\begin{array}{llll}\hfill \mathrm{cos}\alpha & =\frac{\phantom{\rule{-0.17em}{0ex}}\left[4,1\right]\cdot \phantom{\rule{-0.17em}{0ex}}\left[2,-3\right]}{\phantom{\rule{-0.17em}{0ex}}|\phantom{\rule{-0.17em}{0ex}}\left[4,1\right]|\cdot \phantom{\rule{-0.17em}{0ex}}|\phantom{\rule{-0.17em}{0ex}}\left[2,-3\right]|}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\frac{8-3}{\sqrt{{4}^{2}+{1}^{2}}\cdot \sqrt{{2}^{2}+\phantom{\rule{-0.17em}{0ex}}{\left(-3\right)}^{2}}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\frac{5}{\sqrt{17}\cdot \sqrt{13}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \approx \frac{5}{14,86}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \approx 0,336.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Dermed er vinkelen

 $\alpha ={\mathrm{cos}}^{-1}\phantom{\rule{-0.17em}{0ex}}\left(0,336\right)\approx 70,3\text{°}.$

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