Linear Algebra

Cross Product

Two vectors span a patch of plane. The cross product is that patch's area, pointed the way the patch faces. Each component is the area of the patch's shadow on one wall.

aba × b

Drag to orbit · shift-drag to pan · scroll to zoom · set the vectors with the sliders

Cross product—

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Three shadows, three components

Each card looks straight at one wall. The signed area of the patch's shadow there is one component of a × b. Hover a card to light up its wall in the scene.

yz plane—

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zx plane—

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xy plane—

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Scenario

Vector a

x 1.50

y 1.30

z 0.00

Vector b

x 0.40

y -0.50

z 1.50

Show the shadows on the wallsSpin the scene

Values & calculations

|a × b|, the patch's area—

|a| |b| sin θ—

angle between—

a · (a × b)—

b · (a × b)—

The unit vectors multiply in the cycle x → y → z → x. Going with the cycle gives +, against it −, and anything with itself spans no area. Rows are a, columns are b.
×	x̂	ŷ	ẑ
x̂	0	+ẑ	−ŷ
ŷ	−ẑ	0	+x̂
ẑ	+ŷ	−x̂	0

What to observe

Bring the arrows into line: the patch between them collapses and the whole vector a × b goes to zero. An area wearing a direction has nothing left to point.
Each component is the signed area of the patch's shadow on one wall. The cards show each shadow head-on next to a_pb_q − a_qb_p, and the two always agree.
Lay the patch flat on the floor: the yz and zx shadows are just lines, so only the z component survives and a × b points straight up.
Press Swap: every shadow is traced the other way round, so every component flips sign while the area stays put.

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