We have: #A = <4, 9, 1># and #B = <5, 8, - 3>#
First, let's determine #C#:
#=> C = A - B#
#=> C = <4, 9, 1> - <5, 8, - 3>#
#=> C = <(4 - 5), (9 - 8), (1 - (- 3))>#
#=> C = <- 1, 1, 4>#
Then, let's determine the angle between vectors #A# and #C#.
The dot product between two vectors is given as #A cdot B = abs(A) abs(B) cos(theta)#.
We can rearrange this to get:
#=> cos(theta) = (A cdot B) / (abs(A) abs(B))#
#=> cos(theta) = (A cdot C) / (abs(A) abs(C))#
#=> cos(theta) = (<4, 9, 1> cdot <- 1, 1, 4>) / (abs(<4, 9, 1>) abs(<- 1, 1, 4>))#
#=> cos(theta) = ((4 cdot (- 1)) + (9 cdot 1) + (1 cdot 4)) / (sqrt(4^(2) + 9^(2) + 1^(2)) cdot sqrt((- 1)^(2) + 1^(2) + 4^(2)))#
#=> cos(theta) = (- 4 + 9 + 4) / (sqrt(42) cdot sqrt(18))#
#=> cos(theta) = (9) / (sqrt(756))#
#=> cos(theta) = (9) / (sqrt(4 cdot 3 cdot 7 cdot 9))#
#=> cos(theta) = (9) / (6 sqrt(21))#
#=> cos(theta) = (3) / (2 sqrt(21))#
#=> theta = arccos((3) / (2 sqrt(21)))#
#=> theta approx 70.9^(circ)#
