Question

If `A = <2 ,1 ,6 >`, `B = <5 ,2 ,3 >` and `C=A-B`, what is the angle between A and C?

Answer 1

If `C=A-B`
Then `C = <2,1,6> - <5,2,3>`
So `C = <-3,-1,3>`

The dot/scalar product states that for two vectors `vec x`and `vec y` (where `vec x = < x_1 , x_2 , x_3 >` and `vec y = < y_1 , y_2 , y_3 >` )
`vec x * vec y` is equal to two things:
1. = `| vec x| |vec y| cos(Theta)`, where `Theta` denotes the angle between the vectors
2. = `x_1y_1+x_2y_2+x_3y_3`

Therefore it is possible to equate these two and solve for `Theta`

It is important to note, if you didn't know so already, that `|vec x| = sqrt(x_1^2 + x_2^2+ x_3^2)`

So 1. `sqrt(2^2+1^2+6^2)sqrt((-3)^2+(-1)^2+3^2)cos(Theta)`
This is equivalent to `sqrt(41)sqrt(19)cos(Theta)`
Which is: `sqrt(779)cos(Theta)`

1. `2xx(-3)+1xx(-1)+6xx3`
   This is just `11`

So therefore: `sqrt(779)cos(Theta)=11`
`Cos(Theta)=11/sqrt(779)`
`Theta = 1.17` radians

Hope this helped.