How to find the angle between two 3-dimensional vectors:

The formula to find the cosine of the angle is: cosA= u.v/|u|x|v|; 1.u.v means that you multiply the x coordinates together, then the y coordinates and the z coordinates, and add them all together: (x₁x₂+y₁y₂+z₁z₂); 2.|u|x|v| means that you have to find the distance from the origin forboth coordinates and times them together: √(x₁²+y₁²+z₁²)√(x₂²+y₂²+z₂²). 3. This means that if vector u is (2,3,4) and vector v is (5,6,7), the cosine of the angle between them willbe:cosA=(2x5+3x6+4x7)/√(2²+3²+4²)x√(5²+6²+7²) = (10+18+28)/√(4+9+16)x√(25+36+49)=56/√29x√110=0.9915; and therefore A= cos^-10.9915= 7.47579˚≈ to 3 s.f.