Chapter 6.3, Problem 17E

(For some background on the cross product in ${ℝ}^n$ , seeExercise 6.2.44.) Consider three linearly independentvectors ${\stackrel{\to }{v}}_1,{\stackrel{\to }{v}}_2,{\stackrel{\to }{v}}_3$ in ${ℝ}^4$ .
a. What is the relationship between $V\left({\stackrel{\to }{v}}_1,{\stackrel{\to }{v}}_2,{\stackrel{\to }{v}}_3\right)$ and $V\left({\stackrel{\to }{v}}_1,{\stackrel{\to }{v}}_2,{\stackrel{\to }{v}}_3\times {\stackrel{\to }{v}}_2\times {\stackrel{\to }{v}}_3\right)$ ? See Definition 6.3.5.Exercise 6.2.44c is helpful.
b. Express $V\left({\stackrel{\to }{v}}_1,{\stackrel{\to }{v}}_2,{\stackrel{\to }{v}}_3\times {\stackrel{\to }{v}}_2\times {\stackrel{\to }{v}}_3\right)$ in terms ofbut $\Vert {\stackrel{\to }{v}}_1\times {\stackrel{\to }{v}}_2\times {\stackrel{\to }{v}}_3\Vert$ .
c. Use parts (a) and (b) to express $V\left({\stackrel{\to }{v}}_1,{\stackrel{\to }{v}}_2,{\stackrel{\to }{v}}_3\right)$ interms of $\Vert {\stackrel{\to }{v}}_1\times {\stackrel{\to }{v}}_2\times {\stackrel{\to }{v}}_3\Vert$ . Is your result still true whenthe ${\stackrel{\to }{v}}_i$ are linearly dependent?
(Note the analogy to the fact that for two vectors ${\stackrel{\to }{v}}_1$ and ${\stackrel{\to }{v}}_2$ in ${ℝ}^3$ , $\Vert {\stackrel{\to }{v}}_1\times {\stackrel{\to }{v}}_2\Vert$ is the area of the parallelogram defined by ${\stackrel{\to }{v}}_1$ and ${\stackrel{\to }{v}}_2$ .)