The Vector Product of Two Vectors

The vector product (or cross product) of two vectors combines them to produce a third vector, not a number — hence the name. Its magnitude equals the product of the two magnitudes and the sine of the angle between them, and its direction is perpendicular to the plane containing both original vectors:

The direction $\hat n$ is fixed by the right-hand (screw) rule: curl the fingers of the right hand from $\vec A$ (first vector) towards $\vec B$ (second vector) through the smaller angle between them; the thumb then points along $\hat n$, the direction of $\vec A\times\vec B$.

Properties:

1. $\vec A\times\vec B$ is always perpendicular to both $\vec A$ and $\vec B$ (i.e. orthogonal to the plane containing them), whether or not $\vec A$ and $\vec B$ are themselves perpendicular to each other.
2. It is not commutative: $\vec A\times\vec B \ne \vec B\times\vec A$; in fact $\vec A\times\vec B = -(\vec B\times\vec A)$ — the two cross products have equal magnitude $AB\sin\theta$ but point in opposite directions.
3. Maximum magnitude when $\sin\theta=1$, i.e. $\theta=90°$ (perpendicular vectors): $|\vec A\times\vec B|_{max}=AB$.
4. Minimum (zero) when $\sin\theta=0$, i.e. $\theta=0°$ or $180°$ — the cross product of two parallel or anti-parallel (non-zero) vectors vanishes.
5. Self-cross-product is always zero: $\vec A\times\vec A = (AA\sin0°)\hat n = \vec 0$; in physics the zero vector is simply written $0$.
6. Consequently $\hat i\times\hat i=\hat j\times\hat j=\hat k\times\hat k=0$.
7. For the orthogonal unit vectors, following the right-hand screw rule: $\hat i\times\hat j=\hat k$, $\hat j\times\hat k=\hat i$, $\hat k\times\hat i=\hat j$; and, since the product is anti-commutative, $\hat j\times\hat i=-\hat k$, $\hat k\times\hat j=-\hat i$, $\hat i\times\hat k=-\hat j$.
8. In component form, using the determinant recipe,

(note the middle, $\hat j$, component has its two terms in the opposite order to the $\hat i$ and $\hat k$ components — a common slip).

9. Area of a parallelogram: if $\vec A$ and $\vec B$ form two adjacent sides of a parallelogram, the parallelogram's area equals $|\vec A\times\vec B|$. …