# If A = <2 ,4 ,-3 >, B = <3 ,4 ,1 > and C=A-B, what is the angle between A and C?

Mar 24, 2018

The angle between $\vec{A} \mathmr{and} \vec{C}$ is ${63.23}^{0}$

#### Explanation:

$\vec{C} = \vec{A} - \vec{B} = \left(< 2 , 4 , - 3 >\right) - \left(< 3 , 4 , 1 >\right)$

$= \left(< \left(2 - 3\right) , \left(4 - 4\right) , \left(- 3 - 1\right) >\right) = < - 1 , 0 , - 4 >$

$\vec{A} = < 2 , 4 , - 3 > \mathmr{and} \vec{C} = < - 1 , 0 , - 4 >$ Let $\theta$ be the

angle between them ; then we know

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

=(2* (-1)+(4*0)+(-3*(-4)))/(sqrt(2^2+4^2+(-3)^2)* (sqrt((-1)^2+0^2+(-4)^2))

$= \frac{10}{\sqrt{29} \cdot \sqrt{17}} = \frac{10}{22.2} \approx 0.4504$

$\therefore \theta = {\cos}^{-} 1 \left(0.4504\right) \approx {63.23}^{0}$

The angle between $\vec{A} \mathmr{and} \vec{C}$ is ${63.23}^{0}$ [Ans]