If #A = <3 ,-1 ,-7 >#, #B = <5 ,6 ,-4 ># and #C=A-B#, what is the angle between A and C?

1 Answer
May 23, 2017

The angle is #=68.7#º

Explanation:

Let's start by calculating

#vecC=vecA-vecB#

#vecC=〈3,-1,-7〉-〈5,6,-4〉=〈-2,-7,-3〉#

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=〈3,-1,-7〉.〈-2,-7,-3〉=-6+7+21=22#

The modulus of #vecA#= #∥〈3,-1,-7〉∥=sqrt(9+1+49)=sqrt59#

The modulus of #vecC#= #∥〈-2,-7,-3〉∥=sqrt(4+49+9)=sqrt62#

So,

#costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=22/(sqrt59*sqrt62)=0.36#

#theta=68.7#º