Calculus 3 — 5.2: Vector Projection and Why Order Matters

Full course — free exercises, Feynman reviews, and AI-graded feedback: https://ludium.ai/courses/calculus-3 The projection formula has two vector slots, and swapping them is the single most common mistake in this topic. The projection of v onto w and the projection of w onto v are completely different vectors. This video builds the full vector projection formula and shows how to keep the order straight every time. Key concepts covered: The vector projection proj_w v = (v · w / |w|²) w as a scalar projection times a unit vector Using |w|² in the denominator, so no square root is needed Why proj of b onto A is not the same as proj of A onto b — the subscript carries the direction Vector projection (a full vector) versus component projection (a signed scalar) A fully worked example with A = (1, 2, 3) and b = (2, -1, 1) ━━━━━━━━━━━━━━━━━━━━━━━━ SOURCE MATERIALS The source materials for this video are from    • Calculus 3 Lecture 11.3:  Using the Dot Pr...