Creating Perpendicular Vectors

Calculus 3 · Axiom Academy

Creating Perpendicular Vectors Two vectors point wherever they like. There's one honest way to build a third vector that is square to both at once — and that machine is the cross product. The one direction both vectors agree to avoid Give me two vectors and that aren't parallel. They pin down a flat plane. Now ask for a vector that is perpendicular to both of them at the same time — square to and square to . Up to length, there is exactly one such direction: straight out of that plane. The cross product is the rule that hands it to you. Watch and settle into the ground plane and fill in the parallelogram they span. Then lifts straight up out of that plane — and the running dot products and hold at exactly 0 the whole way, the signature of "perpendicular to both." Both dot products stay pinned at zero — that is what " is perpendicular to both" means, written in arithmetic. Its length measures how much the two vectors disagree The direction is settled — but how long is ? Swing around and watch: the length is , which is exactly the area of the parallelogram and span. It peaks when they're perpendicular and collapses to 0 when they line up — parallel vectors span no area, so they make no perpendicular. here, so the length maxes at 15 (at ) and drops to 0 when they're parallel. Two ways out of the plane — the order picks one

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