Chapter 6.2, Problem 44E

The cross product in ${ℝ}^n$ . Consider the vectors ${\stackrel{\to }{v}}_2,{\stackrel{\to }{v}}_3,\mathrm{...},{\stackrel{\to }{v}}_n$ in ${ℝ}^n$ . The transformation
$T\left(\stackrel{\to }{x}\right)=\det \left[\begin{array}{ccccc}|& |& |& & |\\ \stackrel{\to }{x}& {\stackrel{\to }{v}}_2& {\stackrel{\to }{v}}_3& \cdots & {\stackrel{\to }{v}}_n\\ |& |& |& & |\end{array}\right]$ is linear. Therefore, there exists a unique vector $\stackrel{\to }{u}$ in ${ℝ}^n$ such that
$T\left(\stackrel{\to }{x}\right)=\stackrel{\to }{x}\cdot \stackrel{\to }{u}$ for all $\stackrel{\to }{x}$ in ${ℝ}^n$ . Compare this with Exercise 2.1 .43c. This vector $\stackrel{\to }{u}$ is called the cross product of ${\stackrel{\to }{v}}_2,{\stackrel{\to }{v}}_3,\mathrm{...},{\stackrel{\to }{v}}_n$ , written as $\stackrel{\to }{u}={\stackrel{\to }{v}}_2\times {\stackrel{\to }{v}}_3\times \cdots \times {\stackrel{\to }{v}}_n$ .
In other words, the cross product is defined by the fact that

$\begin{array}{l}\stackrel{\to }{x}\cdot \left({\stackrel{\to }{v}}_2\times {\stackrel{\to }{v}}_3\times \cdots \times {\stackrel{\to }{v}}_n\right)\\ =\det \left[\begin{array}{ccccc}|& |& |& & |\\ \stackrel{\to }{x}& {\stackrel{\to }{v}}_2& {\stackrel{\to }{v}}_3& \cdots & {\stackrel{\to }{v}}_n\\ |& |& |& & |\end{array}\right],\end{array}$
for all $\stackrel{\to }{x}$ in ${ℝ}^n$ . Note that the cross product in ${ℝ}^n$ is defined for $n-1$ vectors only. (For example, you can not form the cross product of just two vectors in ${ℝ}^4$ .) Since the ith component of a vector $\stackrel{\to }{w}$ is $\stackrel{\to }{e}{\cdot }_i\stackrel{\to }{w}$ . we can find the cross product by components as follows:
ith component of ${\stackrel{\to }{v}}_2\times {\stackrel{\to }{v}}_3\times \cdots \times {\stackrel{\to }{v}}_n$
   $\begin{array}{l}={\stackrel{\to }{e}}_i\cdot \left({\stackrel{\to }{v}}_2\times \cdots \times {\stackrel{\to }{v}}_n\right)\\ =\det \left[\begin{array}{ccccc}|& |& |& & |\\ {\stackrel{\to }{e}}_i& {\stackrel{\to }{v}}_2& {\stackrel{\to }{v}}_3& \cdots & {\stackrel{\to }{v}}_n\\ |& |& |& & |\end{array}\right]\end{array}$
a. When is ${\stackrel{\to }{v}}_2\times {\stackrel{\to }{v}}_3\times \cdots \times {\stackrel{\to }{v}}_n=\stackrel{\to }{0}$ ? Give your answer in terms of linear independence.
b. Find ${\stackrel{\to }{e}}_2\times {\stackrel{\to }{e}}_3\times \cdots \times {\stackrel{\to }{e}}_n$ .
c. Show that ${\stackrel{\to }{v}}_2\times {\stackrel{\to }{v}}_3\times \cdots \times {\stackrel{\to }{v}}_n$ is orthogonal to all the vectors ${\stackrel{\to }{v}}_i$ , for $i=2,\mathrm{...},n$ .
d. What is the relationship between ${\stackrel{\to }{v}}_2\times {\stackrel{\to }{v}}_3\times \cdots \times {\stackrel{\to }{v}}_n$ and ${\stackrel{\to }{v}}_3\times {\stackrel{\to }{v}}_2\times \cdots \times {\stackrel{\to }{v}}_n$ ?(We swap the first two factors.)
e. Express $\det \left[{\stackrel{\to }{v}}_2\times {\stackrel{\to }{v}}_3\times \cdots \times {\stackrel{\to }{v}}_n{\stackrel{\to }{v}}_2\times {\stackrel{\to }{v}}_3\times \cdots \times {\stackrel{\to }{v}}_n\right]$ in terms of $\Vert {\stackrel{\to }{v}}_2\times {\stackrel{\to }{v}}_3\times \cdots \times {\stackrel{\to }{v}}_n\Vert$ .
f. How do we know that the cross product of two vectors in ${ℝ}^3$ , as defined here. is the same as the standard cross product in ${ℝ}^3$ ? See Definition A.9 of the Appendix.