Cross Product Magnitude Calculator
Compute the cross product a × b of two 3D vectors, its magnitude, and the area of the parallelogram they span.
How it works
a × b = (a_y b_z − a_z b_y, a_z b_x − a_x b_z, a_x b_y − a_y b_x) · |a × b| = √(x² + y² + z²)
The cross product of two 3D vectors is a new vector perpendicular to both, whose length equals |a||b|·sin θ. That length is the area of the parallelogram spanned by a and b, so half of it is the area of the triangle they form. When the vectors are parallel the cross product is the zero vector and its magnitude is zero.
Worked example
For a = (1, 2, 3) and b = (4, 5, 6): a × b = (2·6 − 3·5, 3·4 − 1·6, 1·5 − 2·4) = (−3, 6, −3). Its magnitude is √(9 + 36 + 9) = √54 ≈ 7.3485, which is the area of the parallelogram they span.
Sources: Cross product (Wikipedia)
Frequently asked questions
What does the magnitude of the cross product represent geometrically?
It is the area of the parallelogram formed by the two vectors, equal to |a||b|·sin θ. Halving it gives the area of the triangle with those two vectors as sides.
When is the cross product zero?
When the two vectors are parallel or anti-parallel (or one is the zero vector). Then sin θ = 0, so the parallelogram collapses and the cross product is the zero vector with magnitude 0.
Is the cross product commutative?
No — it is anti-commutative: b × a = −(a × b). Swapping the order reverses the direction of the result, though the magnitude stays the same.
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This calculator is for educational purposes. Double-check important results before acting on them.