Vector Cross Product

The vector cross product (or simply vector product, as opposed to the scalar product (which is also called the dot product, or inner product)) is commonly used in vector calculus--a basic mathematical toolset used in mechanics [272,260], acoustics [352], electromagnetism [359], quantum mechanics, and more. It can be defined symbolically in the form of a matrix determinant:^B.19

$$\underline{x}\times \underline{y} = \left\vert \begin{array}{ccc} \underline{e}_1 & \underline{e}_2 & \underline{e}_3\\ [2pt] x_1 & x_2 & x_3\\ [2pt] y_1 & y_2 & y_3 \end{array}\right\vert$$

$$= (x_2 y_3 - y_2 x_3)\underline{e}_1 + (x_3y_1 - y_3x_1) \underline{e}_2 + (x_1y_2- y_1x_2) \underline{e}_3$$

$$= \left[\begin{array}{ccc} 0 & -x_3 & x_2\\ [2pt] x_3 & 0 & -x_1\\ [2pt] -x_2 & x_1 & 0 \end{array}\right] \left[\begin{array}{c} y_1 \\ [2pt] y_2 \\ [2pt] y_3\end{array}\right] \eqsp \left[\begin{array}{ccc} 0 & y_3 & -y_2\\ [2pt] -y_3 & 0 & y_1\\ [2pt] y_2 & -y_1 & 0 \end{array}\right] \left[\begin{array}{c} x_1 \\ [2pt] x_2 \\ [2pt] x_3\end{array}\right]$$

$$\protect$$ (B.15)

where $\underline{e}_i$ denote the unit vectors in $\mathbb{R}^3$ . The cross-product is a vector in 3D that is orthogonal to the plane spanned by $\underline {x}$ and $\underline{y}$ , and is oriented positively according to the right-hand rule.^B.20

The second and third lines of Eq.(B.15) make it clear that $y\times x = -\; x\times y$ . This is one example of a host of identities that one learns in vector calculus and its applications.