What Is the 3-D Cross Product?

Given two three-dimensional vectors $\vec v$ and $\vec w$, the cross product of $\vec v$ and $\vec w$, denoted by $\vec v\times \vec w$, is defined as the vector that is perpendicular to both $\vec v$ and $\vec w$, and whose magnitude is $|\vec v||\vec w| \sin\theta$. Note that the cross product is a vector, not a real number.

Note that there are actually two vectors that satisfy the above condition. In order to maintain a consistent definition, we use the right hand rule: $\vec v\times \vec w$ is the direction your thumb points to if your index finger is $\vec v$ and middle finger is $\vec w$ in your right hand.

The cross product satisfies many properties.

• The cross product is anticommutative: $a\times b=-(b\times a)$.
• If two vectors $a,b$ are parallel, then the angle between them is $\theta=0$. Thus, $\sin\theta=0$, so $a\times b=0$.
• The cross product distributes over addition.
• The cross product satisfies Jacobi's Identity: $$a\times(b\times c)+b\times(c\times c)+c\times(a\times b)=0$$

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The 2-D "cross product" is actually better understood in 3-D.