Vector product

Definition — Vector product

$\vec u\times\vec v = (u_2 v_3 - u_3 v_2,\; u_3 v_1 - u_1 v_3,\; u_1 v_2 - u_2 v_1).$

• Modulus: $|\vec u\times\vec v| = |\vec u|\,|\vec v|\sin\theta$.
• Direction: perpendicular to the plane of $\vec u$ and $\vec v$, with sense given by the right-hand rule.
• Anticommutative: $\vec u\times\vec v = -\,\vec v\times\vec u$.

Unlike the scalar product, the vector product returns a vector: it is the tool for constructing a direction perpendicular to two given vectors.

Property — Parallelism and scalar triple product

• $\vec u\times\vec v = \vec 0 \iff \vec u\parallel\vec v$.
• The scalar triple product $\vec u\cdot(\vec v\times\vec w)$ gives the (signed) volume of the parallelepiped built on $\vec u,\vec v,\vec w$.

Example

Let $\vec u=(1;-2;1)$ and $\vec v=(-1;1;3)$. Then $\vec u\times\vec v = (-2\cdot 3-1\cdot 1,\; 1\cdot(-1)-1\cdot 3,\; 1\cdot 1-(-2)(-1)) = (-7;-4;-1).$ Perpendicularity check: $(-7;-4;-1)\cdot(1;-2;1) = -7+8-1 = 0$  ✓.

Links

Topics: Analytic geometry in space
Concepts: Scalar triple product · Vector product · Vector
Skills: Analytic geometry · Using formulae