Orthogonal Projection

w = proj_Vw + proj_V^⊥w

Plane V = span{e₁, e₂}

e₁—

e₂—

e₁, e₂ are an orthonormal basis of V (‖e₁‖ = ‖e₂‖ = 1, e₁ ⊥ e₂), so proj_V(w) = ⟨w, e₁⟩ e₁ + ⟨w, e₂⟩ e₂.

Vector w (not in V)

w—

Decomposition

Legend

e₁ (basis of V)

e₂ (basis of V)

⟨w, e₁⟩ e₁

⟨w, e₂⟩ e₂

proj_V(w)

proj_V⊥(w)

V⊥ axis / construction

plane V

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Axes labelled x, y, z