# If A = <2 ,4 ,-1 >, B = <-1 ,8 ,-5 > and C=A-B, what is the angle between A and C?

Mar 3, 2017

The angle is $= 118.5$º

#### Explanation:

Let's start by calculating

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

vecC=〈2,4,-1〉-〈-1,8,-5〉=〈3,-4,4〉

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=〈2,4,-1〉.〈3,-4,4〉=6-16-4=-14

The modulus of $\vec{A}$= ∥〈2,4,-1〉∥=sqrt(4+16+1)=sqrt21

The modulus of $\vec{C}$= ∥〈3,-4,4〉∥=sqrt(9+16+16)=sqrt41

So,

costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=-14/(sqrt21*sqrt41)=-0.48

$\theta = 118.5$º