2D Vector Cross Product
Compute the scalar magnitude of the cross product of two 2D vectors U and V. Use it to find the signed area of the parallelogram they span or to test if two vectors are parallel.
About this calculator
In 3D, the cross product of two vectors produces a new vector perpendicular to both. When both vectors lie in the 2D xy-plane (with zero z-components), the 3D cross product's x and y components vanish, leaving only a single scalar z-component: U × V = u1·v2 − u2·v1. This scalar represents the z-component of the resulting 3D vector and equals |U|·|V|·sin(θ), where θ is the angle between the vectors. Its absolute value equals the area of the parallelogram formed by U and V. The sign indicates orientation: positive means V is counterclockwise from U, negative means V is clockwise. If the result is zero, the vectors are parallel or one is a zero vector.
How to use
Let U = (3, 2) and V = (1, 4). Enter u1 = 3, u2 = 2, v1 = 1, v2 = 4. Apply the formula: U × V = (u1·v2) − (u2·v1) = (3·4) − (2·1) = 12 − 2 = 10. The scalar cross product is 10. This also means the parallelogram formed by U and V has an area of |10| = 10 square units, and V is oriented counterclockwise relative to U.
Frequently asked questions
What does the sign of the 2D cross product tell you about two vectors?
A positive cross product (u1·v2 − u2·v1 > 0) means vector V is counterclockwise from vector U when viewed from the standard orientation. A negative result means V is clockwise from U. A result of zero means the vectors are parallel or anti-parallel, lying along the same line. This signed orientation test is heavily used in computational geometry, for example to determine whether a point lies to the left or right of a directed line segment.
How is the 2D cross product used to calculate the area of a triangle or parallelogram?
The absolute value of u1·v2 − u2·v1 gives the area of the parallelogram spanned by vectors U and V. Dividing by 2 gives the area of the triangle with those vectors as two of its sides. This is one of the fastest ways to compute triangle area from coordinates without needing the base and height explicitly. It is widely used in graphics rendering, mesh computations, and geographic information systems for area calculations from vertex coordinates.
Why is the cross product of two 2D vectors a scalar and not a vector?
Strictly speaking, the cross product is a 3D operation. When two 2D vectors are treated as 3D vectors with z = 0, their cross product has x = 0 and y = 0, leaving only the z-component: u1·v2 − u2·v1. Since this z-component is the only non-trivial result, it is reported as a single scalar in 2D contexts. This scalar can be thought of as the magnitude of the perpendicular vector pointing out of the xy-plane, positive for counterclockwise orientation and negative for clockwise.