# If A = <7 ,-5 ,6 >, B = <4 ,4 ,9 > and C=A-B, what is the angle between A and C?

Dec 13, 2016

The angle is $= 62.6$º

#### Explanation:

We start by calculating $\vec{C}$

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

=〈7,-5,6〉-〈4,4,9〉=〈3,-9,-3〉

The angle between 2 vectors is given by the dot product.

vecA.vecC=∥vecA∥*∥vecC∥*costheta

The dot product is 〈7,-5,6〉.〈3,-9,-3〉=21+45-18=48

The modulus of vecA=∥〈7,-5,6〉∥=sqrt(49+25+36)=sqrt110

The modulus of vecC=∥〈3,-9,-3〉∥=sqrt(9+81+9)=sqrt99

costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)

$= \frac{48}{\sqrt{110} \sqrt{99}} = 0.46$

$\theta = 62.6$º