# If A = <4 ,2 ,5 >, B = <-8 ,3 ,6 > and C=A-B, what is the angle between A and C?

Feb 10, 2016

#### Explanation:

The angle between 2 vectors$\underline{a} \textcolor{b l a c k}{\text{ and }} \underline{b}$ can be found using :

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

where $\theta \textcolor{b l a c k}{\text{ is the angle between the vectors }}$

here C = A - B = (4,2,5) - (-8,3,6) = (12,-1,-1)

$\underline{a} . \underline{c} = \left(4 , 2 , 5\right) . \left(12 , - 1 , - 1\right) = 48 - 2 - 5 = 41$

$| \underline{a} | = \sqrt{{4}^{2} + {2}^{2} + {5}^{2}} = \sqrt{45}$

and $| \underline{c} | = \sqrt{{12}^{2} + {\left(- 1\right)}^{2} + {\left(- 1\right)}^{2}} = \sqrt{146}$

 rArr costheta = 41/(sqrt45 xx sqrt146

and $\theta = {\cos}^{-} 1 \left(\frac{41}{\sqrt{45} . \sqrt{146}}\right) = 1.04 \textcolor{b l a c k}{\text{ radians }}$