A worked example: (1,2,3) × (4,5,6)
The cross product of (1,2,3) and (4,5,6) is (−3, 6, −3), a new vector with a magnitude of 7.348469 — the area of the parallelogram that the two original vectors span.
The cross product formula
For a = (a₁,a₂,a₃) and b = (b₁,b₂,b₃), the cross product is a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁). Applied to (1,2,3) × (4,5,6): the first component is (2×6 − 3×5) = 12−15 = −3, the second is (3×4 − 1×6) = 12−6 = 6, and the third is (1×5 − 2×4) = 5−8 = −3 — giving (−3, 6, −3), matching the calculator's result exactly.
Why the result is a vector, not a number
Unlike the dot product, which collapses two vectors into a single scalar, the cross product's result is itself a new vector — perpendicular to both original vectors — which is why it's only defined in 3D (and, in a specialized generalized form, 7D): perpendicular to two vectors is only a uniquely determined direction in exactly those dimensions.
The special case of perpendicular unit vectors
Crossing the two perpendicular unit vectors (1,0,0) and (0,1,0) gives (0,0,1) with a magnitude of exactly 1 — since the vectors are already perpendicular and both have length 1, the parallelogram they span is a unit square, and its area (the cross product's magnitude) is exactly 1.
Order matters — it's anti-commutative
Swapping the order flips the sign of every component: b × a = −(a × b), the reverse direction of a × b. This is different from the dot product, where a·b and b·a always give the identical number regardless of order.