# If A = <4 ,3 ,-7 >, B = <5 ,7 ,-3 > and C=A-B, what is the angle between A and C?

Dec 24, 2016

The angle is =70.8º

#### Explanation:

Let's calculate $\vec{C}$

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

vecC=〈4,3,-7〉-〈5,7,-3〉=〈-1,-4,-4〉

The angle between the $\vec{A}$ and $\vec{C}$ is given by the dot product.

vecA.vecC=∥vecA∥*∥vecC∥*costheta

The dot product is

vecA.vecC=〈4,3,-7〉.〈-1,-4,-4〉=(-4-12+28)=12

The modulus of $\vec{A}$ is =∥vecA∥=∥〈4,3,-7〉∥=sqrt(16+9+49)=sqrt74

The modulus of $\vec{C}$ is =∥vecC∥=∥〈-1,-4,-4〉∥=sqrt(1+1+16)=sqrt18

Therefore,

costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=12/(sqrt74sqrt18)=0.329

theta=70.8º