# If A = <4 ,-5 ,-4 >, B = <-9 ,1 ,-6 > and C=A-B, what is the angle between A and C?

Feb 26, 2016

$0.801 \text{ radians} \mathmr{and} {45.67}^{\circ}$

#### Explanation:

If $A = < \textcolor{b l u e}{4 , - 5 , - 4} >$
and $B = < \textcolor{g r e e n}{- 9 , 1 , - 6} >$
then
$\textcolor{w h i t e}{\text{XXX}} C = A - B = < \textcolor{b l u e}{4} - \left(\textcolor{g r e e n}{- 9}\right) , \textcolor{b l u e}{- 5} - \textcolor{g r e e n}{1} , \textcolor{b l u e}{- 4} - \left(\textcolor{g r e e n}{- 6}\right) >$

$\textcolor{w h i t e}{\text{XXXX}} = < \textcolor{red}{13 , - 6 , 2} >$

${\theta}_{A C} = \arccos \left(\frac{A \cdot C}{| | A | | \times | | C | |}\right)$

$A \cdot C = \textcolor{b l u e}{4} \times \textcolor{red}{13} + \textcolor{b l u e}{\left(- 5\right)} \times \textcolor{red}{\left(- 6\right)} + \textcolor{b l u e}{\left(- 4\right)} \times \textcolor{red}{2}$
$\textcolor{w h i t e}{\text{XXX}} = 76$

$| | A | | = \sqrt{{\textcolor{b l u e}{4}}^{2} + {\textcolor{b l u e}{\left(- 5\right)}}^{2} + {\textcolor{b l u e}{\left(- 4\right)}}^{2}}$
$\textcolor{w h i t e}{\text{XXX}} = \sqrt{57}$

$| | C | | = \sqrt{{\textcolor{red}{13}}^{2} + {\textcolor{red}{\left(- 6\right)}}^{2} + {\textcolor{red}{2}}^{2}}$
$\textcolor{w h i t e}{\text{XXX}} = \sqrt{209}$

$\theta = \arccos \left(\frac{76}{\sqrt{57} \cdot \sqrt{209}}\right) = \arccos \left(\frac{4}{\sqrt{33}}\right)$

color(white)("X")=0.801 " radians" or 45.67^@