Here is the parallelogram P hovering in midair Let M N and O

Here is the parallelogram P hovering in midair. Let M, N, and O be its orthogonal projections (=shadows) onto the xy-plane, the xz-plane and the yz-plane, respectively.

Prove that if m is the area of M, n that of N, o that of O, and p that of P, then

m² + n² + o² = p².

Start your proof by letting u = (u₁,u₂,u₃) and v = (v₁,v₂,v₃) be vectors that span the parallelogram P.

Solution

https://drive.google.com/file/d/0B2XEvx_GW13beFh0M1QwQmV5bVk/view?usp=sharing

https://drive.google.com/file/d/0B2XEvx_GW13bcE8yTlJhdmxiZGM/view?usp=sharing

https://drive.google.com/open?id=0B2XEvx_GW13bam1qZkxpbUFEdmM

Please go to the links to find the images of the solution. I have uploaded it there.