Vector or Cross Product

It combines two 3D vectors to produce a third, perpendicular vector. The length of that new vector equals the area of the parallelogram the two input vectors span—meaning it reaches a maximum when the inputs meet at a right angle, and drops to the zero vector when they are parallel or point in opposite directions.

Because two opposite directions are perpendicular to any flat plane, the direction is chosen by the right-hand rule: point the fingers of your right hand along the first vector, curl them toward the second, and your thumb points in the direction of a ⨯ b. Swapping the order flips the result upside down (a ⨯ b = −b ⨯ a), so order always matters.

In physics and engineering, ⨯ shows up wherever rotation or magnetism is at work. Torque (rotational force) is the cross product of a lever arm and an applied force, and the magnetic Lorentz force on a moving electric charge is the cross product of its velocity and the magnetic field. Paired with the del operator (∇ ⨯ F), it writes the curl of a vector field, which measures how much a fluid or field swirls around a point.

Is ⨯ the same as the multiplication sign ×?

On paper they look alike, but for computers they are distinct. The ordinary multiplication sign × (U+00D7) lives in the basic Latin supplement and is used everywhere from arithmetic to screen dimensions. The operator ⨯ (U+2A2F) was added to fill out technical math sets so that formulas could distinguish vector cross products from ordinary multiplication in semantic markup such as MathML.

In practice, even most university textbooks, web pages, and LaTeX documents (\times) output the standard × for cross products because it is easier to type and supported by every font. If you are searching a document or writing code that parses formulas, keep in mind that either character might be present.

Where did this notation come from?

It grew out of nineteenth-century work on three-dimensional algebra. William Rowan Hamilton's quaternions (an extension of complex numbers to four dimensions, introduced in 1843) and Hermann Grassmann's algebra of extension (which combined geometric quantities) laid the mathematical foundations. In the 1880s, Josiah Willard Gibbs and Oliver Heaviside independently separated the scalar and vector parts of quaternion multiplication into the dot and cross products used in physics today.

Gibbs called it the "skew product" and wrote it with a cross; Edwin Bidwell Wilson's 1901 textbook Vector Analysis, based on Gibbs's lectures, helped make the cross notation standard worldwide.