Cross Product

The cross product of three-dimensional vectors a = $\left(\vphantom{
a_{1},a_{2},a_{3}}\right.$a1, a2, a3$\left.\vphantom{
a_{1},a_{2},a_{3}}\right)$ and b = $\left(\vphantom{ b_{1},b_{2},b_{3}}\right.$b1, b2, b3$\left.\vphantom{ b_{1},b_{2},b_{3}}\right)$ is defined by

a×b = $\displaystyle \left(\vphantom{
a_{2}b_{3}-a_{3}b_{2},a_{3}b_{1}-a_{1}b_{3},a_{1}b_{2}-a_{2}b_{1}}\right.$a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1$\displaystyle \left.\vphantom{
a_{2}b_{3}-a_{3}b_{2},a_{3}b_{1}-a_{1}b_{3},a_{1}b_{2}-a_{2}b_{1}}\right)$

To enter the cross symbol, click the cross on the Common Symbols toolbar or in the drop-down panel below itbpF0.3009in0.3009in0.0701inbinop.wmf.

For the following examples, use the vectors a = $\left[\vphantom{
\begin{array}{ccc}
1 & 2 & 3
\end{array}
}\right.$$\begin{array}{ccc}
1 & 2 & 3
\end{array}$$\left.\vphantom{
\begin{array}{ccc}
1 & 2 & 3
\end{array}
}\right]$, b = $\left(\vphantom{
\begin{array}{r}
1 \\
0 \\
-1
\end{array}
}\right.$$\begin{array}{r}
1 \\
0 \\
-1
\end{array}$$\left.\vphantom{
\begin{array}{r}
1 \\
0 \\
-1
\end{array}
}\right)$, c = [3, 2, 1], and d = (2, - 1, 0), as defined previously.

$\blacktriangleright$ Evaluate

a×b = $\left(\vphantom{ -2,4,-2}\right.$ -2, 4, - 2$\left.\vphantom{ -2,4,-2}\right)$         a×c = $\left(\vphantom{ -4,8,-4}\right.$ -4, 8, - 4$\left.\vphantom{ -4,8,-4}\right)$         c×d = $\left(\vphantom{ 1,2,-7}\right.$1, 2, - 7$\left.\vphantom{ 1,2,-7}\right)$


$\left(\vphantom{
\begin{array}{r}
.35 \\
-.73 \\
1.2
\end{array}
}\right.$$\begin{array}{r}
.35 \\
-.73 \\
1.2
\end{array}$$\left.\vphantom{
\begin{array}{r}
.35 \\
-.73 \\
1.2
\end{array}
}\right)$×$\left(\vphantom{
\begin{array}{r}
.85 \\
.32 \\
-.77
\end{array}
}\right.$$\begin{array}{r}
.85 \\
.32 \\
-.77
\end{array}$$\left.\vphantom{
\begin{array}{r}
.85 \\
.32 \\
-.77
\end{array}
}\right)$ = $\left(\vphantom{
\begin{array}{c}
.\,\allowbreak 178\,1 \\
1.\,\allowbreak 289\,5 \\
.\,\allowbreak 732\,5
\end{array}
}\right.$$\begin{array}{c}
.\,\allowbreak 178\,1 \\
1.\,\allowbreak 289\,5 \\
.\,\allowbreak 732\,5
\end{array}$$\left.\vphantom{
\begin{array}{c}
.\,\allowbreak 178\,1 \\
1.\,\allowbreak 289\,5 \\
.\,\allowbreak 732\,5
\end{array}
}\right)$         $\left[\vphantom{
\begin{array}{ccc}
1 & -2 & 5
\end{array}
}\right.$$\begin{array}{ccc}
1 & -2 & 5
\end{array}$$\left.\vphantom{
\begin{array}{ccc}
1 & -2 & 5
\end{array}
}\right]$×$\left[\vphantom{
\begin{array}{ccc}
5 & 3 & -5
\end{array}
}\right.$$\begin{array}{ccc}
5 & 3 & -5
\end{array}$$\left.\vphantom{
\begin{array}{ccc}
5 & 3 & -5
\end{array}
}\right]$ = $\left[\vphantom{
\begin{array}{ccc}
-5 & 30 & 13
\end{array}
}\right.$$\begin{array}{ccc}
-5 & 30 & 13
\end{array}$$\left.\vphantom{
\begin{array}{ccc}
-5 & 30 & 13
\end{array}
}\right]$



Subsections