# If A = <4 ,9 ,-7 >, B = <5 ,8 ,-3 > and C=A-B, what is the angle between A and C?

Nov 22, 2017

The angle between $\vec{A} \mathmr{and} \vec{C}$ is ${49.93}^{0}$

#### Explanation:

$\vec{C} = \vec{A} - \vec{B} = \left(< 4 , 9 , - 7 >\right) - \left(< 5 , 8 , - 3 >\right)$

$= \left(< 4 - 5 , 9 - 8 , - 7 + 3 >\right) = < - 1 , 1 , - 4 >$

vecA =<4,9, -7> and vecC =<-1,1,-4> ; theta be the

angle between them ; then we know

$\cos \theta = \frac{\vec{A} \cdot \vec{C}}{| | \vec{A} | | \cdot | | \vec{C} | |}$

=((4.-1)+(9*1)+(-7*-4))/(sqrt(4^2+9^2+(-7)^2)* (sqrt((-1)^2+1^2+(-4)^2))

$= \frac{33}{\sqrt{146} \cdot \sqrt{18}} = \frac{33}{51.26} \approx 0.6437$

$\therefore \theta = {\cos}^{-} 1 \left(0.6437\right) \approx {49.93}^{0}$

The angle between $\vec{A} \mathmr{and} \vec{C}$ is ${49.93}^{0}$ [Ans]