If #A = <7 ,-3 ,5 >#, #B = <4 ,-7 ,-5 ># and #C=A-B#, what is the angle between A and C?

1 Answer
Jul 13, 2018

The angle is #=54.6^@#

Explanation:

Start by calculating

#vecC=vecA-vecB#

#vecC=〈7,-3,5〉-〈4,-7,-5〉=〈3,4,10〉#

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,-3,5〉.〈3,4,10〉=21-12+50=59#

The modulus of #vecA#= #∥〈7,-3,5〉∥=sqrt(49+9+25)=sqrt83#

The modulus of #vecC#= #∥〈3,4,10〉∥=sqrt(9+16+100)=sqrt125#

So,

#costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=59/(sqrt83*sqrt125)=0.58#

#theta=arccos(0.58)=54.6^@#