If #A = <3 ,-1 ,5 >#, #B = <5 ,2 ,8 ># and #C=A-B#, what is the angle between A and C?

1 Answer
Jan 24, 2017

The angle is #=130.4#º

Explanation:

Let's start by calculating

#vecC=vecA-vecB#

#vecC=〈3,-1,5〉-〈5,2,8〉=〈-2,-3,-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,-1,5〉.〈-2,-3,-3〉=-6+3-15=-18#

The modulus of #vecA#= #∥〈3,-1,5〉∥=sqrt(9+1+25)=sqrt35#

The modulus of #vecC#= #∥〈-2,-3,-3〉∥=sqrt(4+9+9)=sqrt22#

So,

#costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=-18/(sqrt35*sqrt22)=-0.65#

#theta=130.4#º