# What is the angle between <-4 , 0 , 7 >  and  < 8 , -5 , 0 > ?

Feb 12, 2016

${114.88}^{\circ}$

#### Explanation:

We may use the Euclidean inner product of 2 vectors $A \mathmr{and} B$ in ${\mathbb{R}}^{n}$ to find the angle $\theta$ between them as follows :

$A \cdot B = | | A | | | | B | | \cos \theta$.

So in this case we get :

$\theta = {\cos}^{- 1} \left(\frac{A \cdot B}{| | A | | | | B | |}\right)$

$= {\cos}^{- 1} \left(\frac{- 32 + 0 + 0}{\left(\sqrt{65}\right) \left(\sqrt{89}\right)}\right)$

$= {114.88}^{\circ}$.

Note that we could also have used the vector cross product since we work in ${R}^{3}$ and would of got the same answer.
$A \times B = | | A | | | | B | | \sin \theta$.