Vectors

Different from scalars, vectors are arrows with magnitude. The direction that pointed by the arrow is called head.

Since we are familiar with sum and dot product of vectors, we skip them here.

Cross Product

For vectors ${\displaystyle u=<u_1,u_2,u_3>}$ and ${\displaystyle v=<v_1,v_2,v_3>}$, we have ${\displaystyle u\times v=<u_2{v}_3-u_3{v}_2,u_3{v}_1-u_1{v}_3,u_1{v}_2-u_2{v}_1>}$, of course that is also a vector.

It is difficult to remember. But from the determinants of matrices, we can easily derive the cross product.

We write ${\displaystyle u}$ and ${\displaystyle v}$ in the form of matrix, ${\displaystyle u\times v=\left[\begin{array}{ccc}i& j& k\\ u_1& u_2& u_3\\ v_1& v_2& v_3\end{array}\right]}$, and the output ${\displaystyle \left|\begin{array}{cc}{u}_2& u_3\\ v_2& v_3\end{array}\right|i-\left|\begin{array}{cc}{u}_1& u_3\\ v_1& v_3\end{array}\right|j+\left|\begin{array}{cc}{u}_1& u_2\\ v_1& v_2\end{array}\right|k}$, the determinant of the matrix, is actually the vector ${\displaystyle <u_2{v}_3-u_3{v}_2,u_3{v}_1-u_1{v}_3,u_1{v}_2-u_2{v}_1>}$ while ${\displaystyle i,j,k}$ are unit vectors in ${\displaystyle x,y,z}$ directions.

There are some important properties of the cross product of vectors.

• The cross product is a vector perpendicular to both ${\displaystyle u}$ and ${\displaystyle v}$.
• • ${\displaystyle u}$ and ${\displaystyle v}$ in three-space are parallel if and only is ${\displaystyle u\times v=\mathbf{0}}$.
• The magnitude of the cross product is ${\displaystyle |u||v|\sin \theta }$, where ${\displaystyle \sin \theta }$ is the angle between ${\displaystyle u}$ and ${\displaystyle v}$.
• The magnitude of the cross product is the area of the parallelogram with two vectors.
• The volum of the parallelepiped determined by three vectors ${\displaystyle a,b,c}$ in three-space is ${\displaystyle |a\cdot (b\times c)|}$.

And remember that the order of the vectors in their product can not be changed without adding negative sign.