# If A = <2 ,4 ,-3 >, B = <3 ,4 ,-5 > and C=A-B, what is the angle between A and C?

Jan 25, 2017

The angle is $= 131.6$º

#### Explanation:

Let's start by calculating

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

vecC=〈2,4,-3〉-〈3,4,-5〉=〈-1,0,2〉

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,-3〉.〈-1,0,2〉=-2+0-6=-8

The modulus of $\vec{A}$= ∥〈2,4,-3〉∥=sqrt(4+16+9)=sqrt29

The modulus of $\vec{C}$= ∥〈-1,0,2〉∥=sqrt(1+0+4)=sqrt5

So,

costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=-8/(sqrt29*sqrt5)=-0.664

$\theta = 131.6$º