Verify the following identity of Lagrange v times w middot v

Verify the following identity of Lagrange: (v times w) middot (v times w) + (v middot w)^2 =|v||^2| w|^2

Solution

We know that (v x w) = |v||w| sin where is the angle between the vectors u and v. Also, (v.w)= |v||w|cos . Then (v x w).(v x w) + (v.w)² =|v|²|w|²sin²+|v|²|w|²cos² =|v|²|w|²( sin² + cos²)= =|v|²|w|² (as sin² + cos² =1)