two vector \( u \) and \( v \)
\(
u\times v
=
\begin{vmatrix}
i & j & k \\
u_1 & u_2 & u_3 \\
v_1 & v_2 & v_3 \\
\end{vmatrix}
\)
is perpendicular to \( u \) and \( v \).
The cross product \( v\times u \) is \( -(u\times v) \)
\(
u\cdot (u\times v)
=
v\cdot (u\times v)
=
0
\)
\(
u\times u = v\times v = 0
\)
\(
\lVert u\times v\rVert
=
\lVert u\rVert
\lVert v\rVert
|\sin\theta|
\)
\(
|u\cdot v|
=
\lVert u\rVert
\lVert v\rVert
|\cos\theta|
\)
Triple Product
\(
(u\times v)\cdot w
=
\begin{vmatrix}
u_1 & u_2 & u_3 \\
v_1 & v_2 & v_3 \\
w_1 & w_2 & w_3 \\
\end{vmatrix}
=
\begin{vmatrix}
w_1 & w_2 & w_3 \\
u_1 & u_2 & u_3 \\
v_1 & v_2 & v_3 \\
\end{vmatrix}
=
\text{Volumn}
\)
If
\(
(u\times v)\cdot w
=0
\) that means \( u,v,w \) lies in same plane.