Question

If `A = <7 ,9 ,4 >`, `B = <6 ,-9 ,-3 >` and `C=A-B`, what is the angle between A and C?

Answer 1

`theta = cos^-1((197)/(sqrt(146) sqrt(374))) approx 0.568` radians or `32.54^circ`

Explanation:

`vec c = [7-6, 9-(-9), 4-(-3)] = [1, 18, 7]`

The angle between vectors is given by:

`theta =cos^-1((vec a cdot vec c)/(||vec a|| ||vec c||))`

Calculating individual parts:
`vec a cdot vec c = 7*1 + 9*18 + 4*7=197`
`||vec a || = sqrt(7^2 +9^2 + 4^2) = sqrt(146)`
`||vec c || = sqrt(1^2 +18^2 + 7^2) = sqrt(374)`

So:

`theta =cos^-1((197)/(sqrt(146) sqrt(374)))`, which is about as good as it gets, but approximations are:

`theta approx 0.568` radians or `approx 32.54^circ`