Dot Product

Calculus 2 · Axiom Academy

One number, two ways to find it — and its sign tells you the angle between the vectors. 1. Multiply Matching Components, Then Add For and , the dot product is built from the components directly: pair up the first entries, pair up the second entries, multiply each pair, and add. The result is a single number. multiply matching components, then sum 2. The Same Number, Geometrically There is a second way to get the very same number. Using the vectors' lengths and the angle between them: the component sum and are always equal Watch stay put while swings around it. Both readouts are computed live — one from 's components, one from the lengths and the measured angle — and they stay locked together at every instant. Small means near 1 , so the product is large and positive — the vectors strongly agree in direction. As grows the product shrinks; past a right angle turns negative and so does . The geometric form is what lets the dot product recover the angle. Solving for gives — one dot product and two lengths, and the angle drops out. 3. The Sign Tells You the Angle Because and are always positive, the sign of is the sign of — and that pins down the angle. Sweep all the way around the fixed and the sign reads off the geometry directly. — acute. The vectors lean the same way. — obtuse. The vectors lean apart. One scalar, found two ways — and its sign hands you the angle between the vectors. Scroll up to revisit any step.

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