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

Mar 14, 2018

The angle is $= {36.1}^{\circ}$

#### Explanation:

Start by calculating

$\vec{C} = \vec{A} - \vec{B}$

vecC=〈7,-5,4〉-〈4,7,-3〉=〈3,-12,7〉

The angle between $\vec{A}$ and $\vec{C}$ is given by the dot product definition.

vecA.vecC=∥vecA∥*∥vecC∥costheta

Where $\theta$ is the angle between $\vec{A}$ and $\vec{C}$

The dot product is

vecA.vecC=〈7,-5,4〉.〈3,-12,7〉=21+60+28=109

The modulus of $\vec{A}$= ∥〈7,-5,4〉∥=sqrt(49+25+16)=sqrt90

The modulus of $\vec{C}$= ∥〈3,-12,7〉∥=sqrt(9+144+49)=sqrt202

So,

costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=109/(sqrt90*sqrt202)=0.81

$\theta = \arccos \left(0.81\right) = {36.1}^{\circ}$