What is the angle between <7,-3,1 >  and < -2,8,-5 >?

Feb 25, 2016

$2.19 \text{ radians } \left({125.5}^{\circ}\right)$

Explanation:

To calculate the angle between 2 vectors $\underline{a} \text{ and " ulb " use}$

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

let $\underline{a} = \left(7 , - 3 , 1\right) \text{ and } \underline{b} = \left(- 2 , 8 , - 5\right)$

(1) calculate the$\textcolor{b l u e}{\text{ the dot product "ula . ulb }}$

$\underline{a} . \underline{b} = \left(7 , - 3 , 1\right) . \left(- 2 , 8 , - 5\right)$

$= \left(7 \times - 2\right) + \left(- 3 \times 8\right) + \left(1 \times - 5\right) = - 14 - 24 - 5 = - 43$

(2) calculate the$\textcolor{b l u e}{\text{ magnitudes of }} \underline{a} , \underline{b}$

$| \underline{a} | = \sqrt{{7}^{2} + {\left(- 3\right)}^{2} + {1}^{2}} = \sqrt{49 + 9 + 1} = \sqrt{59}$

$| \underline{b} | = \sqrt{{\left(- 2\right)}^{2} + {8}^{2} + {\left(- 5\right)}^{2}} = \sqrt{4 + 64 + 25} = \sqrt{93}$

$\Rightarrow \theta = {\cos}^{-} 1 \left(\frac{- 43}{\sqrt{59} \times \sqrt{93}}\right) = 2.19 \text{ radians }$