If #A = <-7 ,1 ,5 >#, #B = <3 ,6 ,-2 ># and #C=A-B#, what is the angle between A and C?

1 Answer
Jun 18, 2017

The angle is #=28.9#º

Explanation:

Let's start by calculating

#vecC=vecA-vecB#

#vecC=〈-7,1,5〉-〈3,6,-2〉=〈-10,-5,7〉#

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=〈-7,1,5〉.〈-10,-5,7〉=70-5+35=100#

The modulus of #vecA#= #∥〈-7,1,5〉∥=sqrt(49+1+25)=sqrt75#

The modulus of #vecC#= #∥〈-10,-5,7〉∥=sqrt(100+25+49)=sqrt174#

So,

#costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=100/(sqrt75*sqrt174)=0.88#

#theta=28.9#º