Cross Product

A common operation between two ${\mathbb{R}}^3$ vectors. $u\times v=(u_2{v}_3-u_3{v}_2,u_3{v}_1-u_1{v}_3,u_1{v}_2-u_2{v}_1)$ $u\times v=||v||||w||\sin \theta$

Cross Product as a Matrix

$$\det \left[\begin{array}{ccc}i& j& k\\ a_1& a_2& a_3\\ b_1& b_2& b_3\end{array}\right]=i\left[\begin{array}{cc}{a}_2& a_3\\ b_2& b_3\end{array}\right]-j\left[\begin{array}{cc}{a}_1& a_3\\ b_1& b_3\end{array}\right]+k\left[\begin{array}{cc}{a}_1& a_2\\ b_1& b_2\end{array}\right]$$

Properties

• Length of $a\times b$ is an area of the parllelogram spanned by $a$ and $b$
• $||a\times b||=||a||||b||\sin \theta$
• $a\times b$ orthogonal to $a$ and $b$
• Linearity : $(au+bv)\times w=a(u\times w)+b(v\times w)$
• Anticommutivity : $(u\times v)=-(v\times u)$
• Dependence detecting : $\{u,v\}$ dep $⟹u\times v=0$
• Distributivity under product: $a\times (b+c)=a\times b+a\times c$
• Distributivity under addition: $(a+b)\times c=a\times c+a\times b$