# What is the angle between <-1,-7,2 >  and <-3,-5,1> ?

Mar 6, 2016

${23}^{\circ}$

#### Explanation:

An easy way to find the angle $\theta$ between any 2 vectors $A \mathmr{and} B$ in ${\mathbb{R}}^{3}$ is from the Euclidean inner product :

$\cos \theta = \frac{A \cdot B}{| | A | | | | B | |}$

$\therefore \theta = {\cos}^{- 1} \left(\frac{\left(- 1 , - 7 , 2\right) \cdot \left(- 3 , - 5 , 1\right)}{| | \left(- 1 , - 7 , 2\right) | | | | \left(- 3 , - 5 , 1\right)}\right)$

$= {\cos}^{- 1} \left(\frac{3 + 35 + 2}{\sqrt{1 + 49 + 4} \sqrt{9 + 25 + 1}}\right)$

$= {23}^{\circ}$.

(An alternative would be to use the vector cross product but this method would take longer : $\sin \theta = \frac{A \times B}{| | A | | | | B | |}$.