If #A = <6 ,4 ,-3 >#, #B = <7 ,1 ,-4 ># and #C=A-B#, what is the angle between A and C?

1 Answer

The angle between A and C is #cos^-1(3/sqrt(671))#.

Explanation:

Let us consider A=(6,4,-3) into a vector #vec A=6hati+4hatj-3hatk# and similarly with B as #vecB=7hati+hatj-4hatk#.
For C=A-B as given above for this,
The difference of the vectors #vecA# and #vecB# is given by
#vecA-vecB=(a_1-b_1)hati+(a_2-b_2)hatj+(a_3-b_3)hatk#

Substituting the values and calculating we get,
#:.C=-hati+3hatj+hatk#
So, now we can find the angle between the A and C and this is given by,
#theta=cos^-1{{a_1b_1+a_2b_2+a_3b_3}/{sqrt((a_1)^2+(a_2)^2+(a_3)^2)sqrt((b_1)^2+(b_2)^2+(b_3)^2)]}#

substituting and calculating we get,

the angle between the A and C is #theta = cos^-1(3/sqrt(671))#.