The magnitude of the cross product is the area of a parallelogram
The cross product A×B of two vectors A and B is interestingly just "a single number" (in two dimensions), and its absolute value is exactly equal to the area of the parallelogram formed by A and B. What is more, its sign tells you whether B is clockwise or counterclockwise from A. Drag the arrows to see for yourself.
Formula: When A=(a₁,a₂) and B=(b₁,b₂), A×B = a₁b₂ − a₂b₁. If this value is positive, B is counterclockwise from A; if it is negative, B is clockwise from A.
💡 In three dimensions, the cross product is not a number but a new vector (its direction is perpendicular to both vectors and is determined by the right-hand rule). This concept is essential in physics for calculating torque and the direction of magnetic fields.