# If A = <2 ,4 ,-7 >, B = <3 ,5 ,2 > and C=A-B, what is the angle between A and C?

Apr 30, 2018

The angle between $\vec{A} \mathmr{and} \vec{C}$ is ${41.13}^{0}$

#### Explanation:

$\vec{C} = \vec{A} - \vec{B} = \left(< 2 , 4 , - 7 >\right) - \left(< 3 , 5 , 2 >\right)$

$= \left(< 2 - 3 , 4 - 5 , - 7 - 2 >\right) = < - 1 , - 1 , - 9 >$

vec A =<2,4, -7> and vec C =<-1,-1,-9> ; theta

be the angle between them ; then we know

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

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

$= \frac{57}{\sqrt{69} \cdot \sqrt{83}} = \frac{57}{75.68} . \approx 0.7532$

$\therefore \theta = {\cos}^{-} 1 \left(0.7532\right) \approx {41.13}^{0}$

The angle between $\vec{A} \mathmr{and} \vec{C}$ is ${41.13}^{0}$ [Ans]