Pseudo-scalar product

The pseudo-scalar product (also called the vector product in the plane or the 2D cross product) of two vectors in the plane is a number that measures how much the two vectors 'turn' with respect to each other. For vectors $\vec{a} = (a_x, a_y)$ and $\vec{b} = (b_x, b_y)$, the pseudo-scalar product is defined as $a_x b_y - a_y b_x$. This value is positive if $\vec{b}$ is to the left of $\vec{a}$, negative if to the right, and zero if the vectors are parallel.

Important properties

• The pseudo-scalar product is zero if and only if the vectors are parallel.

• Its absolute value equals the area of the parallelogram formed by the two vectors.

• The sign indicates the orientation (left/right turn) from $\vec{a}$ to $\vec{b}$.

• It is anti-symmetric: swapping the vectors changes the sign.