Question

If `A = <8 ,1 ,4 >`, `B = <6 ,5 ,-8 >` and `C=A-B`, what is the angle between A and C?

Answer 1

First figure out the `C`.

`C = A - B = langle8,1,4rangle - langle6,5,-8rangle=langle8-6,1-5,4--8rangle=langle2,-4,12rangle`

So now we need to find the angle between `A` and `C`. We can use this formula `costheta=frac{u*v}{||u||\ ||v||}`.

`A*C = langle8,1,4rangle*langle2,-4,12rangle=8*2+1*-4+4*12=60`

`||A||=sqrt{8^2+1^2+4^2} = sqrt{81}=9`
`||C||=sqrt{2^2+(-4)^2+12^2} = sqrt164 =2sqrt41`

So putting these values together would get you `costheta=frac{60}{9*2sqrt41} = frac{60}{sqrt41} = \frac{10}{3\sqrt41}`

Now take the inverse of cosine function on both sides to obtain `theta = cos^{-1}(frac{10}{3sqrt41})\approx58.6289°`