If #A = <7 ,-5 ,-2 >#, #B = <4 ,-8 ,-3 ># and #C=A-B#, what is the angle between A and C?

1 Answer
Jan 10, 2017

The angle is #=84.04#º

Explanation:

Let's start by calculating

#vecC=vecA-vecB#

#vecC=〈7,-5,-2〉-〈4,-8,-3〉=〈3,3,1〉#

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=〈7,-5,-2〉.〈3,3,1〉=21-15-2=4#

The modulus of #vecA#= #∥〈7,-5,-2〉∥=sqrt(49+25+4)=sqrt78#

The modulus of #vecC#= #∥〈3,3,1〉∥=sqrt(9+9+1)=sqrt19#

So,

#costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=4/(sqrt78*sqrt19)=0.104#

#theta=84.04#º