Open Educational Resources

Vectors and their Operations: Cross product

The cross product and its properties

The cross product of two vectors is an operation that takes two vectors \bold A and \bold B and returns a vector \bold C. This operation is denoted by \bold C =\bold A\times\bold B and is read “C equals A cross B”. The magnitude of  \bold C = \bold A \times \bold B is defined as,

(2.22)   \[|\bold C|=|\bold A||\bold B|\sin\theta,\quad 0\le\theta \le 180^\circ\]

in which \theta is the angle between the two vectors.

The definition of the magnitude of the cross product implies that, \bold C=\bold 0 if two non-zero vectors are parallel.

By definition, \bold C is perpendicular to both \bold A and \bold B, which are not parallel, and its sense of direction is specified by the right-hand rule. In other words, \bold C is perpendicular to the plane containing the non-parallel vectors \bold A and \bold B such that \bold C (or \bold u_C) is in the direction of the thumb following the right-hand rule (Fig. 2.26a). The right-hand rule can also be equivalently demonstrated by the right-hand three-finger rule as demonstrated in Fig. 2.26b.

Fig. 2.26  (a) The right-hand rule and (b) the right-hand three-finger rule for determining the direction of a cross product.

If \bold u_C is the unit vector of \bold C, the resultant of a cross product \bold A\times \bold B can be written as,

(2.23)   \[\bold C = \bold A\times\bold B=|\bold A||\bold B|\sin \theta\ \bold u_C \]

The following interactive tool illustrates \bold C = \bold A\times \bold B. Use the sliders to change the angles of \bold A and \bold B with the Cartesian axes, and observe the resultant of their cross product.

Properties of Cross product
  1. Anti-commutativity: \bold A\times\bold B=-(\bold B\times\bold A)
  2. Associativity: for c being a scalar, c(\bold A\times \bold B)=(c\bold A)\times \bold B=\bold A\times (c\bold B)=(\bold A\times \bold B)c
  3. Distributivity: \bold A\times(\bold B+\bold C)=(\bold A\times\bold B)+(\bold A\times\bold C)

Cross product in CVN

In a Cartesian frame the following relationships are readily obtained for the mutually perpendicular unit vectors \bold i, \bold j and \bold k.

(2.24)   \[ \begin{matrix} \bold i\times \bold i=\bold 0 &\bold i\times \bold j=\bold k & \bold i\times \bold k=-\bold j \\ \bold j\times \bold i=-\bold k & \bold j\times \bold j=\bold 0 & \bold j\times \bold k=\bold i \\ \bold k\times \bold i=\bold j & \bold k\times \bold j=-\bold i & \bold k\times \bold k=\bold 0  \end{matrix} \]

Based on the above results and the properties of the cross product, the cross product of two vectors in their CVN is,

    \[ \begin{split} \bold A&=A_x\bold i + A_y\bold j+A_z\bold k\quad , \quad \bold B=B_x\bold i+B_y\bold j+B_z\bold k\\\bold C = \bold A\times \bold B&=(A_xB_x)\bold i\times \bold i +(A_xB_y)\bold i\times \bold j +(A_xB_z)\bold i\times \bold k\\&+(A_yB_x)\bold j\times \bold i +(A_yB_y)\bold j\times \bold j +(A_yB_z)\bold j\times \bold k \\&+(A_zB_x)\bold k\times \bold i +(A_zB_y)\bold k\times \bold j +(A_zB_z)\bold k\times \bold k \end{split}\]

leading to,

(2.25)   \[ \bold C = \bold A\times\bold B=(A_yB_z-A_zB_y)\bold i+(A_zB_x-A_xB_z)\bold j+(A_xB_y-A_yB_x)\bold k \]

This result need not to be memorized and can be obtained by calculating the following determinant symbolically.

(2.26)   \[\bold A\times\bold B = \left\|\begin{matrix} \bold i & \bold j & \bold k \\ A_x & A_y & A_z \\ B_x & B_y & B_z \\ \end{matrix}\right\|\]

As a reminder, the determinant of a two-by-two matrix M is a scalar defined as,

    \[M=\begin{bmatrix}a && b\\ c && d\end{bmatrix}\implies\|M\|=ad-bc\]

Consequently, the determinant of a three-by-three matrix M is a scalar defined as

    \[M=\begin{bmatrix}a && b && c\\ d && e && f \\ g && h && i\end{bmatrix}\implies\|M\|=a\left\|\begin{matrix} e & f \\ h & i \end{matrix}\right\| - b\left\|\begin{matrix} d & f \\ g & i \end{matrix}\right\| + c\left\|\begin{matrix} d & e \\ g & h \end{matrix}\right\|\]

The following tool calculates the cross product \bold C=\bold A\times \bold B. The vector \bold C is drawn in green.