Computing Dot Products

Pre-Calculus · Axiom Academy

EXAMPLE Computing Dot Products Use the dot product to turn two vectors into one number — and read the angle between them straight out of it. Find the angle between the vectors and . Along the way you will compute their dot product and magnitudes. Both vectors drawn from the origin: u = ⟨1, 1⟩ points up-right at 45°, v = ⟨0, 2⟩ points straight up. The arc marks the angle θ we want. Nice work — you computed a dot product and used it to find the exact angle between two vectors. Dot product: multiply matching components and add, . The result is a single number (a scalar), here . Perpendicular shortcut: a dot product of 0 means , so the vectors meet at — a one-line test for perpendicularity. The dot product is the bridge between algebra and geometry: a quick component calculation tells you how two vectors are oriented relative to each other.

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