Vector (Cross) Product of Two Vectors

The second way of multiplying two vectors, the vector product (or cross product), of

$\vec A$ and $\vec B$ is defined as

where $\theta$ is the angle between $\vec A$ and $\vec B$, and $\hat n$ is a unit vector

perpendicular to the plane containing both $\vec A$ and $\vec B$. Unlike the dot product,

the result of a cross product is itself a vector.

Direction — the right-hand rule. The direction of $\hat n$ (and hence of

$\vec A \times \vec B$) is found by curling the fingers of the right hand from $\vec A$

towards $\vec B$ through the smaller angle $\theta$ between them; the extended thumb then

points along $\vec A \times \vec B$ (see the figure). Because of this rule, the cross

product is anticommutative: reversing the order of multiplication reverses the direction

of the result, $\vec A \times \vec B = -(\vec B \times \vec A)$.

Geometric meaning. The magnitude $|\vec A \times \vec B| = AB\sin\theta$ is numerically

equal to the area of the parallelogram formed by $\vec A$ and $\vec B$ as its two adjacent

sides (base $\times$ height, with $B\sin\theta$ as the perpendicular height).

Component formula. For two vectors lying entirely in the xy-plane,

$\vec A = A_x\hat i + A_y\hat j$ and $\vec B = B_x\hat i + B_y\hat j$, the cross product has

only a z-component (perpendicular to the plane), given by

Special cases. When $\vec A$ and $\vec B$ are parallel ($\theta = 0^\circ$), the cross

product is zero (no parallelogram area is enclosed). When they are perpendicular

($\theta = 90^\circ$), the magnitude is at its maximum, $AB$. Physical applications: …