T TsAmE Junior Member Joined Aug 28, 2010 Messages 55 Sep 9, 2010 #1 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.
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.
S soroban Elite Member Joined Jan 28, 2005 Messages 5,584 Sep 9, 2010 #2 Hello, TsAmE! \(\displaystyle \text{Show that: }\;\text{orth}_{\bf a}(\bf b) \;=\; b - \text{proj}_{a}(b)\) Click to expand... A diagram should make it clear . . . Code: * * | b * | * | orth_a(b) * | * | * - - - - - - - - * - - - * proj_a(b) : - - - - - - a - - - - - : \(\displaystyle \text{We see that: }\;\text{proj}_{\bf a}(\bf b) + \text{orth}_a(b) \;=\; b\) \(\displaystyle \text{Therefore: }\;\text{orth}_{\bf a}(\bf b) \;=\; b - \text{proj}_a(b)\)
Hello, TsAmE! \(\displaystyle \text{Show that: }\;\text{orth}_{\bf a}(\bf b) \;=\; b - \text{proj}_{a}(b)\) Click to expand... A diagram should make it clear . . . Code: * * | b * | * | orth_a(b) * | * | * - - - - - - - - * - - - * proj_a(b) : - - - - - - a - - - - - : \(\displaystyle \text{We see that: }\;\text{proj}_{\bf a}(\bf b) + \text{orth}_a(b) \;=\; b\) \(\displaystyle \text{Therefore: }\;\text{orth}_{\bf a}(\bf b) \;=\; b - \text{proj}_a(b)\)
