# What is the angle between <7 , 2 , 7 >  and  < 8 , -5 , 0 > ?

##### 1 Answer
Jan 18, 2016

${61.132}^{\circ}$.

#### Explanation:

The angle $\theta$ between any 2 vectors $A \mathmr{and} B$ in ${\mathbb{R}}^{n}$ may be found using the Euclidean inner product as follows :

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

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

$= {\cos}^{- 1} \left(\frac{7 \times 8 + 2 \times - 5 + 7 \times 0}{\sqrt{{7}^{2} + {2}^{2} + {7}^{2}} \sqrt{{8}^{2} + {5}^{2} + {0}^{2}}}\right)$

$= {\cos}^{- 1} \left(\frac{46}{\sqrt{102} \sqrt{89}}\right)$

$= {61.132}^{\circ}$.

Alternatively, since these particular vectors are in ${\mathbb{R}}^{3}$, we could also have used the vector cross product, where $A \times B = | | A | | | | B | | \sin \theta$.
This will involve a matrix determinant to evaluate and I leave the details as an exercise. It will eventually also give the same final answer.