The Dot Product

Linear Algebra (Matrices) · Axiom Academy

One number that measures how much two vectors point the same way — and it equals . The dot product has two faces. Algebraically , pair matching components, multiply, and add. Geometrically , multiply the two lengths by the cosine of the angle between them. They are not two operations — they are the same number, computed two ways. Algebraic: pair, multiply, add Geometric: lengths times the cosine of the angle 2. Alignment Is What It Measures Hold fixed and swing around. The dot product is largest when they aim the same way, slides to zero the instant they are perpendicular, and goes negative as swings past, bottoming out when they point opposite. Watch rise, vanish, and fall. , so hits its maximum . Here that is . , so exactly — the turning point between positive and negative. is the signed shadow of along . The dot product is times that shadow. The cosine carries the sign, so the dot product alone tells direction at a glance: means an acute angle (pointing similarly), means an obtuse angle (pointing apart). For example , give , so leans away from . The most useful special case is a dot product of zero . Because and the lengths are positive, the only way to get zero is — that is, . So a single arithmetic check decides a geometric question. Two non-zero vectors are orthogonal (perpendicular) exactly when their dot product is zero — no angle measuring required.

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.