If #A = <2 ,4 ,-9 >#, #B = <3 ,0 ,-5 ># and #C=A-B#, what is the angle between A and C?

1 Answer
Feb 25, 2017

The angle is #30#º

Explanation:

Let's start by calculating

#vecC=vecA-vecB#

#vecC=〈2,4,-9〉-〈3,0,-5〉=〈-1,4,-4〉#

The angle between #vecA# and #vecC# is given by the dot product definition.

#vecA.vecC=∥vecA∥*∥vecC∥costheta#

Where #theta# is the angle between #vecA# and #vecC#

The dot product is

#vecA.vecC=〈2,4,-9〉.〈-1,4,-4〉=-2+16+36=50#

The modulus of #vecA#= #∥〈2,4,-9〉∥=sqrt(4+16+81)=sqrt101#

The modulus of #vecC#= #∥〈-1,4,-4〉∥=sqrt(1+16+16)=sqrt33#

So,

#costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=50/(sqrt101*sqrt33)=0.866#

#theta=30#º