Projections

Pre-Calculus · Axiom Academy

How one vector casts a shadow onto another — and why only part of a force does work. 1. The Shadow of One Vector on Another Picture light shining straight down onto the line of vector . Vector casts a shadow on that line: drop a perpendicular from the tip of , and the segment it lands on is the projection. Two quantities come out of this one picture. A number — the signed length of the shadow A vector — the shadow itself, lying along 2. The Scalar Projection Is a Signed Length The scalar projection is one number: the length of the shadow, with a sign. Because , dividing by leaves a result that depends only on and the angle . , so the scalar projection is positive — the shadow points the same way as . , so the scalar projection is 0 — casts no shadow on at all. , so the scalar projection is negative — the shadow points opposite to . The sign records direction along . A negative value is a real shadow, just on the far side of the origin. 3. The Projection Vector Splits u in Two Multiply the unit vector by the scalar projection and you get the projection vector — the shadow with both length and direction. It splits into a part along and a part perpendicular to it. This leg lies on the line of and is the shadow we just built. What is left over after removing the shadow; it meets at a right angle. Push a box with a force while it slides through a displacement . Only the part of along moves the box — exactly the projection of onto . That is why work is a dot product.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.