If #A = <4 ,-5 ,1 >#, #B = <5 ,8 ,-3 ># and #C=A-B#, what is the angle between A and C?

1 Answer
Jul 27, 2017

The angle is #=42.7^@#

Explanation:

Let's start by calculating

#vecC=vecA-vecB#

#vecC=〈4,-5,1〉-〈5,8,-3〉=〈-1,-13,4〉#

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=〈4,-5,1〉.〈-1,-13,4〉=-4+65+4=65#

The modulus of #vecA#= #∥〈4,-5,1〉∥=sqrt(16+25+1)=sqrt42#

The modulus of #vecC#= #∥〈-1,-13,4〉∥=sqrt(1+169+16)=sqrt186#

So,

#costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=65/(sqrt42*sqrt186)=0.74#

#theta=42.7^@#