Compute the cross product of two 3D vectors, with magnitude, dot product and the angle between them.
The cross product is perpendicular to both input vectors (right-hand rule). Its length equals |A| × |B| × sin(angle), the area of the parallelogram spanned by A and B.
cross product
How to use this tool
Type its x, y and z components.
Type the second vector's components.
The resulting vector, perpendicular to both inputs, is shown big.
Magnitude is the parallelogram area; the angle comes from the dot product.
Formula & worked example
For A = (1, 0, 0) and B = (0, 1, 0): A × B = (0, 0, 1), magnitude 1, dot product 0 and angle 90°.
Limitations — read this before you rely on it
The cross product only exists in 3 dimensions (and, with care, 7). For 2D vectors, pad z with 0. A zero result means the vectors are parallel (or one is the zero vector).
Frequently asked questions
A vector perpendicular to two 3D input vectors, with length equal to the parallelogram area they span.
The unit vectors i = (1,0,0) and j = (0,1,0) give i × j = k = (0,0,1).
The dot product is a number measuring alignment; the cross product is a vector measuring perpendicular area. Dot 0 means perpendicular; cross 0 means parallel.
Follow the right-hand rule: point fingers from A toward B and your thumb points along A × B.
The vectors are parallel (or one is zero), so they span no area.