\section{The Cross Product}

\begin{enumerate}[1.]

\item
{\it Give an example of three non-zero vectors ${\bf u}$, ${\bf v}$, ${\bf w}$ such that ${\bf u}\cdot( {\bf v}\times {\bf w})=4$. }

\bigskip

One example is ${\bf u}=4\kk$, ${\bf v}=\ii$, ${\bf w}={\bf j}$. Then ${\bf v}\times {\bf w}=\ii\times\jj=\kk$ and 
${\bf u}\cdot( {\bf v}\times {\bf w})=4$. 

\bigskip


\item {\it Give an example of three non-zero vectors ${\bf u}$, ${\bf v}$, ${\bf w}$, such that ${\bf v}\times {\bf w}\neq 0$ but ${\bf u}\cdot( {\bf v}\times {\bf w})=0$. }

\bigskip


One example is ${\bf u}={\bf v}=\ii$, ${\bf w}={\bf j}$. Then ${\bf v}\times {\bf w}=\ii\times\jj=\kk\neq 0$ and 
${\bf u}\cdot( {\bf v}\times {\bf w})=\ii\cdot \kk =0$. Any other non-zero vectors such that ${\bf v}$ and ${\bf w}$ are not
colinear but ${\bf u, v,w}$ are coplanar would also work.


\bigskip

\item {\it Find the area of the triangle in $\RR^3$ with vertices $(0,1,1)$, $(2,2,1)$, $(1,0,3)$.}


\bigskip

The triangle is spanned by the vectors ${\bf v}=\langle 2,1,0 \rangle$ and 
${\bf w}=\langle 1,-1,2 \rangle$. We have
$$
{\bf v} \times {\bf w} = 
\left| \begin{matrix}
\ii& \jj & \kk \\
2 & 1 & 0 \\
1 & -1 & 2 \\      
            \end{matrix}\right|   
= (2-0)\ii - (4-0)\jj + (-2-1)\kk = 2\ii -4\jj -3\kk.						
 $$
The area of the triangle is $\displaystyle \frac{1}{2}|{\bf v} \times {\bf w}|
= \frac{1}{2}\sqrt{4+16+9} = \frac{\sqrt{29}}{2}$.


 
\bigskip

\item {\it  
Find the volume of the parallelepiped in ${\mathbb R}^3$ spanned by the vectors 
$\ii-\jj+4\kk$, $3\jj-\kk$, $2\ii+\jj$.}


\bigskip

The volume is
$$V=\left|
\left|\begin{matrix}
1& -1& 4\\
0 & 3& -1\\
2& 1& 0\\
            \end{matrix}\right|\right|
 = |1\cdot 1+1\cdot 2+4\cdot(-6)|=|-21|=21
 $$

\end{enumerate}
