Question

What is the angle between `<5 , 5 , 3 >` and `< 4, 9 , 1 >`?

Answer 1

`26.58` degrees

Explanation:

We're going to use the dot product here. For two vectors `vec(u),vec(v) in RR^3` where `vec(u) = (u_1,u_2,u_3) and vec(v) = (v_1,v_2,v_3)` the dot product is given by the following two formulae:

`vec(u)*vec(v) = u_1v_1 + u_2v_2+u_3v_3`

and

`vec(u)*vec(v) = |vec(u)||vec(v)|costheta`

Combining these gives:

`theta = cos^(-1)((u_1v_1 + u_2v_2+u_3v_3)/(|vec(u)||vec(v)|))`

`|vec(u)| = sqrt(5^2+5^2+3^2) = sqrt(59)`

`|vec(v)| = sqrt(4^2+9^2+1^2) = sqrt(98)`

`theta = cos^(-1)((5*4 + 5*9 + 3*1)/(sqrt(59)sqrt(98))) = 26.58` degrees