# If A = <4 ,7 ,2 >, B = <1 ,1 ,-4 > and C=A-B, what is the angle between A and C?

Mar 8, 2017

The angle is =28º

#### Explanation:

Let's start by calculating

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

vecC=〈4,7,2〉-〈1,1,-4〉=〈3,6,6〉

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=〈4,7,2〉.〈3,6,6〉=12+42+12=66

The modulus of $\vec{A}$= ∥〈4,7,2〉∥=sqrt(16+49+4)=sqrt69

The modulus of $\vec{C}$= ∥〈3,6,6〉∥=sqrt(9+36+36)=sqrt81=9

So,

costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=66/(sqrt69*9)=0.88

$\theta = 28$º