2D Cross Product Calculator – Signed Area

In the plane the cross product collapses to a single number: the determinant uₓv_y − u_yvₓ. For u = (2,3) and v = (4,1): 2×1 − 3×4 = 2 − 12 = −10. The absolute value, 10, is the area of the parallelogram the vectors span; the sign records orientation — negative here because v lies clockwise from u. The calculator prints the determinant subtraction, the area, and the orientation verdict (counterclockwise/positive, clockwise/negative, zero/collinear). Triangles halve it: the triangle with those edge vectors has area 5, which is also the shoelace-formula heart for polygon areas from vertex lists. Zero cross product means the vectors are parallel — (3,6) × (1,2) = 6−6 = 0 — the standard collinearity test for checking whether three points (via two edge vectors) lie on one line.

1. 1

   Enter both 2D vectors

   u = (uₓ, u_y) and v = (vₓ, v_y)

2. 2

   Compute the determinant

   uₓv_y − u_yvₓ = 2−12 = −10 is shown

3. 3

   Take the absolute value

   Parallelogram area = 10

4. 4

   Read orientation

   Negative = clockwise; zero = collinear

• 1

  2D cross product is a scalar — the z of the 3D cross product

• 2

  Sign = orientation: positive counterclockwise, negative clockwise

• 3

  Triangle area is half the parallelogram: |u×v|/2

• Subtracting in the wrong order: uₓv_y − u_yvₓ, not the reverse

• Losing the sign and missing the orientation information

• Using the parallelogram area where a triangle area (half) is needed

• Expecting a vector result — in 2D the cross is a signed number