Remove a component before normalising

THE IDEA

Read the example.

Subtracting the projection of v onto nonzero u leaves a component orthogonal to u. If that remainder is nonzero, dividing by its norm produces the next unit direction. The demo rotates the direction used for projection.

v⊥ = v − (uᵀv / uᵀu)u

Use it well

An exactly zero remainder signals linear dependence. In floating-point work, nearly dependent vectors need numerical care; classical Gram–Schmidt is not always the most stable implementation.