# If A = <-7 ,1 ,-3 >, B = <3 ,4 ,1 > and C=A-B, what is the angle between A and C?

Feb 6, 2017

The angle is $= 23.1$º

#### Explanation:

Let's start by calculating

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

vecC=〈-7,1,-3〉-〈3,4,1〉=〈-10,-3,-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=〈-7,1,-3〉.〈-10,-3,-4〉=70-3+12=79

The modulus of $\vec{A}$= ∥〈-7,1,-3〉∥=sqrt(49+1+9)=sqrt59

The modulus of $\vec{C}$= ∥〈-10,-3,-4〉∥=sqrt(100+9+16)=sqrt125

So,

costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=79/(sqrt59*sqrt125)=0.9199

$\theta = 23.1$º