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

1 Answer
Jan 30, 2017

The angle is #=107.5#º

Explanation:

Let's start by calculating

#vecC=vecA-vecB#

#vecC=〈5,2,8〉-〈2,5,3〉=〈3,-3,5〉#

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=〈5,2,8〉.〈3,3,-5〉=15+6-40=-19#

The modulus of #vecA#= #∥〈5,2,8〉∥=sqrt(25+4+64)=sqrt93#

The modulus of #vecC#= #∥〈3,-3,5〉∥=sqrt(9+9+25)=sqrt43#

So,

#costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=-19/(sqrt93*sqrt43)=-0.3#

#theta=107.5#º