Dot Product

Pre-Calculus · Axiom Academy

One operation, two faces: a quick sum of products on paper, and a measure of how much two vectors point the same way. 1. Multiply Matching Components, Then Add Line the two vectors up by component. Multiply the matching pieces — x with x , y with y — and add the results. Two vectors go in; one number comes out. Multiply matching components, then sum 2. It Measures Shared Direction The same number has a geometric face. Drop a shadow of straight down onto the line of — that shadow is the projection . The dot product is the length of times the length of that shadow, which works out to . and — how long each vector is. — how much they point the same way. It dials the product up or down. Both formulas agree: for and , each gives 12 . Rearranged, this is the angle-finder: . For and : , , , so and . 3. The Sign Tells You the Angle Because and are always positive, the sign of the dot product comes entirely from . Watch swing around a fixed : the readout passes through positive, then exactly zero, then negative. Why the sign works: the lengths never go negative, so takes its sign straight from — and cosine is positive below , zero at , negative above it. The headline use: is the cleanest test for a right angle there is — no square roots, no inverse cosine, just a sum of products. You've seen the dot product as a quick sum of products, as a measure of shared direction, and as a one-line test for the angle between two vectors. Scroll up to revisit any step.

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