Area Between Two Vectors 2D Formula

The area between two vectors in two dimensions is given by their cross product.

For example, to calculate the area of the parallelogram between and , take the cross product of the two vectors.

This formula returns the signed area between the two vectors. If the second vector is a counterclockwise rotation from the first vector, the area is positive.

If the second vector is a clockwise rotation from the first vector, the area is negative.

There are a couple ways to arrive at this formula. You can think about the two vectors as living in three dimensions, in which case their cross product returns another vector that is parallel to the plane and whose length is equal to the area between the two vectors.

Alternatively, you can derive the formula by taking the determinant of the matrix whose columns are the same as the vectors. Since the determinant measures the change in area between the basis vectors and the column vectors, this gives us the same formula for the area.

In either case, this version of the formula calculates the signed area between the two vectors. The area is positive if the second vector is a counterclockwise rotation from the first and negative if its a clockwise rotation from the first.

The area between two vectors in three dimensions is given by the magnitude of their cross product.

The 3D formula returns the unsigned area, because the magnitude always returns a positive number.

Cross Product 2D Formula

The cross product of two vectors in 2D is calculated by the determinant of the corresponding 2x2 matrix. Geometrically the cross product corresponds to the signed area of the parallelogram formed between the two vectors.

Cross Product Formula

The cross product of two vectors in three dimensions produces a vector perpendicular to the two vectors and whose length is eual to the area betwee the two vectors.