# What is the angle between <-3,2,6 >  and <5,0,-5> ?

Feb 20, 2016

2.71 radians ≈ ${155.4}^{\circ}$

#### Explanation:

To calculate the angle between 2 vectors $\underline{a} \text{ and} \underline{b}$
use the following formula.

$\cos \theta = \frac{\underline{a} . \underline{b}}{| \underline{a} | | \underline{b} |}$

let $\underline{a} = \left(- 3 , 2 , 6\right) \text{ and } \underline{b} = \left(5 , 0 , - 5\right)$

hence $\underline{a} . \underline{b} = \left(- 3 , 2 , 6\right) . \left(5 , 0 , - 5\right)$

$= \left(- 3 \times 5\right) + \left(2 \times 0\right) + \left(6 \times - 5\right)$

= -15 + 0 - 30 = - 45

$| \underline{a} | = \sqrt{{\left(- 3\right)}^{2} + {2}^{2} + {6}^{2}} = \sqrt{9 + 4 + 36} = \sqrt{49} = 7$

$| \underline{b} | = \sqrt{{5}^{2} + 0 + {\left(- 5\right)}^{2}} = \sqrt{25 + 0 + 25} = \sqrt{50}$

$\Rightarrow \theta = {\cos}^{-} 1 \left(\frac{- 45}{7 \times \sqrt{50}}\right) = 2.71 \text{ radians}$