Cross Product Calculator
Enter two 3D vectors below to instantly get their cross product and magnitude.
How the Cross Product Is Calculated
For vectors a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃), the cross product a × b is: (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁).
Worked example
a = (1, 2, 3), b = (4, 5, 6)
x: (2×6) − (3×5) = 12 − 15 = −3
y: (3×4) − (1×6) = 12 − 6 = 6
z: (1×5) − (2×4) = 5 − 8 = −3
a × b = (−3, 6, −3)
Frequently Asked Questions
What is the cross product of two vectors?
The cross product of two 3D vectors is a new vector that's perpendicular to both of the original vectors, with a magnitude equal to the area of the parallelogram they form. It's only defined in three dimensions (and seven, in more advanced math, but not in two).
How is the cross product different from the dot product?
The dot product of two vectors gives a single number (a scalar) and measures how much the vectors point in the same direction. The cross product gives a whole new vector, perpendicular to both inputs, and measures the area they sweep out together.
Does the order of the vectors matter?
Yes. a × b = −(b × a) — swapping the order flips the direction of the resulting vector, though its length stays the same. This is different from the dot product, where order doesn't matter.
What does it mean if the cross product is the zero vector?
A cross product of (0, 0, 0) means the two original vectors are parallel (or one of them is the zero vector) — there's no unique plane for a perpendicular vector to stand on.
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