Cross Product

Definition

Let

• $\mathbf{u}=(u_1,u_2,u_3)$ vectors in 3-space
• $\mathbf{v}=(v_1,v_2,v_3)$ vectors in 3-space

Then

• We define the cross product $u\times v$ as

$$\begin{array}{rl}u\times v& =(u_2{v}_3-u_3{v}_2,u_3{v}_1-u_1{v}_3,u_1{v}_2-u_2{v}_1)\\ & =\left(\left|\begin{array}{cc}{u}_2& u_3\\ v_2& v_3\end{array}\right|,-\left|\begin{array}{cc}{u}_1& u_3\\ v_1& v_3\end{array}\right|,\left|\begin{array}{cc}{u}_1& u_2\\ v_1& v_2\end{array}\right|\right)\end{array}$$

Tip: Memorization

Look at the determinant form

$$u\times v=\left(\left|\begin{array}{cc}{u}_2& u_3\\ v_2& v_3\end{array}\right|,-\left|\begin{array}{cc}{u}_1& u_3\\ v_1& v_3\end{array}\right|,\left|\begin{array}{cc}{u}_1& u_2\\ v_1& v_2\end{array}\right|\right)$$

Imagine

$$\left[\begin{array}{ccc}{u}_1& u_2& u_3\\ v_1& v_2& v_3\end{array}\right]$$

• For first $u\times v$ component, just exclude column 1.
• Second component, exclude column 2, but negative
• Third component, exclude column 3