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

1 Answer
May 13, 2017

The angle is #=169.61#º

Explanation:

Let's start by calculating

#vecC=vecA-vecB#

#vecC=〈3,2,5〉-〈5,4,8〉=〈-2,-2,-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=〈3,2,5〉.〈-2,-2,-3〉=-6-4-15=-25#

The modulus of #vecA#= #∥〈3,2,5〉∥=sqrt(9+4+25)=sqrt38#

The modulus of #vecC#= #∥〈-2,-2,-3〉∥=sqrt(4+4+9)=sqrt17#

So,

#costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=-25/(sqrt38*sqrt17)=-0.984#

#theta=169.61#º