# What is the angle between < 4 , 1 , -2 >  and  < -9 , -3 , 2 > ?

Mar 12, 2018

The angle is $= {165.4}^{\circ}$

#### Explanation:

The angle between $2$vectors, $\vec{A}$ and $\vec{B}$ is given by the dot product definition.

vecA.vecB=∥vecA∥*∥vecB∥costheta

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

The dot product is

vecA.vecB=〈4,1,-2〉.〈-9,-3,2〉=(4)*(-9)+(1)*(-3)+(-2)*(2)=-43

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

The modulus of $\vec{B}$= ∥〈-9,-3,2〉∥=sqrt(81+9+4)=sqrt94

So,

costheta=(vecA.vecB)/(∥vecA∥*∥vecB∥)=-43/(sqrt21*sqrt94)=-0.97

$\theta = \arccos \left(- 0.97\right) = {165.4}^{\circ}$