Orthogonal projection (vectors)

[TsAmE]

Show that the vector $\displaystyle orth_{a}b = b - proj_{a}b$ (orthogonal projection of b) is orthogonal to a

I have no idea on how to start. I only know the scalar and vector projection formulae.

 

[soroban]

Hello, TsAmE!

$\displaystyle \text{Show that: }\;\text{orth}_{\bf a}(\bf b) \;=\; b - \text{proj}_{a}(b)$

A diagram should make it clear . . .

                        *
                     *  |
              b   *     |
               *        | orth_a(b)
            *           |
         *              |
      * - - - - - - - - * - - - *
           proj_a(b)
      : - - - - - - a - - - - - :

$\displaystyle \text{We see that: }\;\text{proj}_{\bf a}(\bf b) + \text{orth}_a(b) \;=\; b$

$\displaystyle \text{Therefore: }\;\text{orth}_{\bf a}(\bf b) \;=\; b - \text{proj}_a(b)$