Cross Product & 3D Vectors
The dot product of two vectors hands you a number. The cross product hands you a whole new vector — one that stands straight up out of the plane the first two live in. It's how you find "which way is up", spin a wrench or point a surface. Grade 12 Vectors into first-year.
A product that points somewhere new
Two vectors u and v in 3D span a parallelogram. Their cross product u × v is the vector that is perpendicular to both of them — perpendicular to the whole parallelogram — with a length equal to that parallelogram's area:
Spin it 🟠🔵🟢
Drag the scene to rotate the camera. Set u (orange) and v (blue) with the sliders — the green arrow is u × v, always poking out of the shaded parallelogram. Try the presets: î × ĵ = k̂.
Reading the result
Length = area
|u × v| is the area of the parallelogram u and v span, and it equals |u||v| sin θ. It's biggest when the two vectors are perpendicular and shrinks to nothing as they line up — the mirror image of the dot product, which peaks when they're parallel.
Direction = right-hand rule
Point your right hand's fingers along u and curl them toward v; your thumb points along u × v. Swap the order and the answer flips sign: v × u = −(u × v). It's the one product that isn't commutative.
When u and v are parallel, the parallelogram is flat, its area is zero, and so u × v = 0 — the quickest test for "do these two point the same way?" And notice the answer is always at a perfect right angle to both inputs: the dot products u·(u×v) and v·(u×v) are exactly zero.