Mr. Integral
Math notes/Linear Algebra/Orthogonality and Least Squares

Dot Products and Orthogonal Projection — Cheat sheet

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Core rule

p=aTbaTaa,r=b−p,aTr=0,∥b∥2=∥p∥2+∥r∥2.p=\frac{a^Tb}{a^Ta}a,\quad r=b-p,\quad a^Tr=0,\quad \|b\|^2=\|p\|^2+\|r\|^2.

Watch for

The direction vector must be nonzero. Projection onto a line is a vector, not just its scalar coefficient.