Dot Product Properties

Calculus 3 · Axiom Academy

One number, two ways to read it — the sum of matched components, and , the angle between the vectors. 1. Two Definitions, One Number Take and . Multiply matched components and add: . That single number equals — the lengths times the cosine of the angle between them. Algebraic: sum of matched components 2. The Angle Controls the Sign Hold fixed and swing so the angle opens from onward. Since and the lengths never change, the dot product just tracks — largest when the vectors align, exactly zero when they are perpendicular, and negative once they point apart. , so . The vectors point the same general way. , so exactly — the perpendicular case. , so . The vectors lean apart. , so those axes are perpendicular — no picture needed. The one-line orthogonality test To ask "are these perpendicular?" you never measure an angle — you compute one number. If (and neither is the zero vector), then . 3. The Dot Product Is a Shadow Drop straight down onto the line through . The signed length of that shadow is the scalar projection . So the dot product is literally "how much of points along ," scaled by . Read the sign: a positive shadow means leans along ; a shadow of length zero means has no component along at all — the perpendicular case again. Same number, three faces: a component sum, an angle detector, and the length of a shadow are all the one quantity . The projection is a large positive number; the dot product is large and positive.

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