Question

Find the angle between the given vectors to the nearest tenth of a degree.

u = <6, -1>, v = <7, -4>
Can someone explain how to do this?

Answer 1

Use the dot product relation:

$\mathbf u\cdot\mathbf v=\|\mathbf u\|\|\mathbf v\|\cos\theta$

where $\theta$ is the angle between $\mathbf u$ and $\mathbf v$.

You have

$\mathbf u\cdot\mathbf v=6\times7+(-1)\times(-4)=46$
$\|\mathbf u\|=\sqrt{6^2+(-1)^2}=\sqrt{37}$
$\|\mathbf v\|=\sqrt{7^2+(-4)^2}=\sqrt{65}$

So, the angle is given by

$\cos\theta=\dfrac{46}{\sqrt{37}\sqrt{65}}\implies \theta\approx20.3^\circ$