Vector projection formula derivation

Proj_b a = ((a*b)/||b||^2) b
||Proj_b a|| = ||a||cosθ

Proof: We know that a - Proj_b a = c, a vector perpendicular to b, and Proj_b a = kb, where k is some scalar in the real numbers.
Therefore, b*c = 0; b*(a - Proj_b a) = b*(a - kb) = 0
b*a - kb*b = 0; b*a = kb*b; (b*a)/(b*b) = (b*a)/||b||^2 = k
∴ Proj_b a = kb = ((a*b)/||b||^2) b
||Proj_b a|| = ||(||a|| ||b|| cosθ)/||b||^2)b|| = ||a||cosθ

Oh, God! Linear Algebra final exam! Let's study together!

Doomed! I need to study the stuff with lines and plane and you have to find them with vectors properties. 55555555, the teacher doesn't even giva a name to this chapter. =.=

Ok, let's do this! Ohohohohoho~

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