Cross Product in Three-Dimensional Euclidean Space
Definition
The cross product of $\mathbf{x}$ and $\mathbf{y}$ is defined in terms of $\mathbf{x}, \mathbf{y} \in \mathbb{R}^3$.
$$ \begin{align*} \mathbf{x} \times \mathbf{y} =& (x_{2}y_{3} - x_{3}y_{2}, x_{3}y_{1} - x_{1}y_{3}, x_{1}y_{2} - x_{2}y_{1}) \\ =& \det \begin{bmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ x_{1} & x_{2} & x_{3} \\ y_{1} & y_{2} & y_{3} \end{bmatrix} \\ =& \begin{bmatrix} 0 & -x_{3} & x_{2} \\ x_{3} & 0 & -x_{1} \\ -x_{2} & x_{1} & 0 \end{bmatrix} \begin{bmatrix} y_{1} \\ y_{2} \\ y_{3} \end{bmatrix} \end{align*} $$
Explanation
Note that $\mathbf{i} = (1,0,0)$, $ \mathbf{j} = (0,1,0)$, and $\mathbf{k} = (0,0,1)$. Like the dot product, the cross product can also be defined in a more general sense, but in practice, it is typically considered only in three dimensions. This specific definition in three-dimensional space is also known as a vector product, although this term is usually used only when making a strict distinction. The most common application is in physics, where it frequently appears in expressions for torque and the Lorentz force, among others. Its geometric shape can be easily envisioned by recalling the right-hand rule. Here, we introduce several properties of the cross product without proof.
Properties
For $\mathbf{x}, \mathbf{y}, \mathbb{z} \in \mathbb{R}^3$ and $k \in \mathbb{R}$, the following holds:
(1) $\mathbf{x} \times \mathbf{x} = 0$
(2) Anti commutativity: $\mathbf{x} \times \mathbf{y} = -\mathbf{y} \times \mathbf{x} $
(3) $(k \mathbf{x}) \times \mathbf{y} = k (\mathbf{x} \times \mathbf{y}) = \mathbf{x} \times (k \mathbf{y})$
(4) $\mathbf{x} \times ( \mathbf{y}+ \mathbb{z} )= (\mathbf{x} \times \mathbf{y}) + (\mathbf{x} \times \mathbb{z})$
(5) Scalar triple product: $(\mathbf{x} \times \mathbf{y}) \cdot \mathbb{z} = \mathbf{x} \cdot ( \mathbf{y} \times \mathbb{z})$
(6) Vector triple product (bac-cab rule): $\mathbf{x} \times ( \mathbf{y} \times \mathbb{z} ) = (\mathbf{x} \cdot \mathbb{z}) \mathbf{y} - (\mathbf{x} \cdot \mathbf{y}) \mathbb{z} $
(7) $|| \mathbf{x} \cdot \mathbf{y} ||^2 = (\mathbf{x} \cdot \mathbf{x} ) ( \mathbf{y} \cdot \mathbf{y} ) - ( \mathbf{x} \cdot \mathbf{y} )^2$
(8) $|| \mathbf{x} \times \mathbf{y} || = || \mathbf{x} || || \mathbf{y} || \sin{\theta} $
(9) If $\mathbf{x} \times \mathbf{y} \ne \mathbb{0}$, then $\mathbf{x} \times \mathbf{y} $ is perpendicular to both $\mathbf{x}$ and $\mathbf{y}$.
Many properties are counterintuitive because the commutative law does not hold. Work through problems and write them out on paper to become accustomed to them.