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(For some background on the cross product in ${\mathbf{鈩�}}^{\mathbf{n}}$, see Exercise 6.2.44.) Consider three linearly independent vectors $\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{7}}}\mathbf{,}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{2}}}\mathbf{,}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{3}}}$ in ${\mathbf{鈩�}}^{\mathbf{4}}$.
a. What is the relationship between $\mathbf{V}\left(\stackrel{鈫�}{v_1},\stackrel{鈫�}{v_2},\stackrel{鈫�}{v_3}\right)$and $\mathbf{V}\mathbf{(}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{1}}}\mathbf{,}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{2}}}\mathbf{,}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{3}}}\mathbf{,}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{1}}}\mathbf{脳}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{2}}}\mathbf{脳}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{3}}}\mathbf{)}$? See Definition 6.3.5. Exercise is helpful.
b. Express $\mathbf{V}\mathbf{(}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{1}}}\mathbf{,}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{2}}}\mathbf{,}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{3}}}\mathbf{,}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{1}}}\mathbf{脳}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{2}}}\mathbf{脳}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{3}}}\mathbf{)}$in terms ofrole="math" localid="1660118250452" $\left|{\stackrel{鈫�}{v}}_1脳{\stackrel{鈫�}{v}}_2脳{\stackrel{鈫�}{v}}_3\right|$.
c. Use parts (a) and (b) to express $\mathit{V}\left({\stackrel{鈫�}{v}}_1,{\stackrel{鈫�}{v}}_2,{\stackrel{鈫�}{v}}_3\right)$in terms of $||{\stackrel{鈫�}{v}}_1脳{\stackrel{鈫�}{v}}_2脳{\stackrel{鈫�}{v}}_3||$. Is your result still true when the ${\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathit{i}}$ are linearly dependent?
(Note the analogy to the fact that for two vectors ${\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{1}}$ and ${\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{2}}$ in role="math" localid="1660118435758" ${\mathbf{鈩�}}^{\mathbf{3}}||{\stackrel{鈫�}{v}}_1脳{\stackrel{鈫�}{v}}_2||$is the area of the parallelogram defined by${\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{1}}$ and${\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{2}}$.)

Step by step solution

01

To find the relationship. 

a) To find,

$V\left(\stackrel{鈫�}{v_1},\stackrel{鈫�}{v_2},\stackrel{鈫�}{v_3}\right)=||\stackrel{鈫�}{v_1}||||\stackrel{鈫�鈯�}{v_2}||||\stackrel{鈫�鈯�}{v_3}||$

On the other hand,

$$\begin{gathered} V\left(\stackrel{鈫�}{v_1},\stackrel{鈫�}{v_2},\stackrel{鈫�}{v_3},\stackrel{鈫�}{v_1}脳\stackrel{鈫�}{v_2}脳\stackrel{鈫�}{v_3}\right)=||\stackrel{鈫�}{v_1}||||\stackrel{鈫�鈯�}{v_2}||||\stackrel{鈫�鈯�}{v_3}||||{\left(\stackrel{鈫�}{v_1}脳\stackrel{鈫�}{v_2}脳\stackrel{鈫�}{v_3}\right)}^{鈯�}|| \\ V\left(\stackrel{鈫�}{v_1},\stackrel{鈫�}{v_2},\stackrel{鈫�}{v_3},\stackrel{鈫�}{v_1}脳\stackrel{鈫�}{v_2}脳\stackrel{鈫�}{v_3}\right)=V\left(\stackrel{鈫�}{v_1}脳\stackrel{鈫�}{v_2}脳\stackrel{鈫�}{v_3}\right)||{\left(\stackrel{鈫�}{v_1}脳\stackrel{鈫�}{v_2}脳\stackrel{鈫�}{v_3}\right)}^{鈯�}|| \\ V\left(\stackrel{鈫�}{v_1},\stackrel{鈫�}{v_2},\stackrel{鈫�}{v_3},\stackrel{鈫�}{v_1}脳\stackrel{鈫�}{v_2}脳\stackrel{鈫�}{v_3}\right)=V\left(\stackrel{鈫�}{v_1},\stackrel{鈫�}{v_2},\stackrel{鈫�}{v_3}\right)||\left(\stackrel{鈫�}{v_1}脳\stackrel{鈫�}{v_2}脳\stackrel{鈫�}{v_3}\right)|| \end{gathered}$$

02

To express the terms. 

b) To express the terms,

$$\begin{gathered} V\left(\stackrel{鈫�}{v_1},\stackrel{鈫�}{v_2},\stackrel{鈫�}{v_3},\stackrel{鈫�}{v_1}脳\stackrel{鈫�}{v_2}脳\stackrel{鈫�}{v_3}\right)=\left|det\left[\begin{array}{ccc}\stackrel{鈫�}{v_1}& \stackrel{鈫�}{v_2}& \stackrel{鈫�}{v_3}\end{array}\stackrel{鈫�}{v_1}脳\stackrel{鈫�}{v_2}脳\stackrel{鈫�}{v_3}\right]\right| \\ V\left(\stackrel{鈫�}{v_1},\stackrel{鈫�}{v_2},\stackrel{鈫�}{v_3},\stackrel{鈫�}{v_1}脳\stackrel{鈫�}{v_2}脳\stackrel{鈫�}{v_3}\right)=\left|det\left[\stackrel{鈫�}{v_1}脳\stackrel{鈫�}{v_2}脳\stackrel{鈫�}{v_3}\begin{array}{ccc}\stackrel{鈫�}{v_1}& \stackrel{鈫�}{v_2}& \stackrel{鈫�}{v_3}\end{array}\right]\right| \\ V\left(\stackrel{鈫�}{v_1},\stackrel{鈫�}{v_2},\stackrel{鈫�}{v_3},\stackrel{鈫�}{v_1}脳\stackrel{鈫�}{v_2}脳\stackrel{鈫�}{v_3}\right)={||\left(\stackrel{鈫�}{v_1}脳\stackrel{鈫�}{v_2}脳\stackrel{鈫�}{v_3}\right)||}^2 \end{gathered}$$

03

When the v→i are linearly dependent.

c) From parts a. and b., we compute

$$\begin{gathered} V\left(\stackrel{鈫�}{v_1},\stackrel{鈫�}{v_2},\stackrel{鈫�}{v_3}\right)=\frac{V\left(\stackrel{鈫�}{v_1},\stackrel{鈫�}{v_2},\stackrel{鈫�}{v_3},\stackrel{鈫�}{v_1}脳\stackrel{鈫�}{v_2}脳\stackrel{鈫�}{v_3}\right)}{||\left(\stackrel{鈫�}{v_1}脳\stackrel{鈫�}{v_2}脳\stackrel{鈫�}{v_3}\right)||} \\ V\left(\stackrel{鈫�}{v_1},\stackrel{鈫�}{v_2},\stackrel{鈫�}{v_3},\stackrel{鈫�}{v_1}\right)=\frac{{||\left(\stackrel{鈫�}{v_1}脳\stackrel{鈫�}{v_2}脳\stackrel{鈫�}{v_3}\right)||}^2}{||\left(\stackrel{鈫�}{v_1}脳\stackrel{鈫�}{v_2}脳\stackrel{鈫�}{v_3}\right)||} \\ V\left(\stackrel{鈫�}{v_1},\stackrel{鈫�}{v_2},\stackrel{鈫�}{v_3}\right)=||\left(\stackrel{鈫�}{v_1}脳\stackrel{鈫�}{v_2}脳\stackrel{鈫�}{v_3}\right)|| \end{gathered}$$

If $\stackrel{鈫�}{v_1},\stackrel{鈫�}{v_2}$and $\stackrel{鈫�}{v_3}$are linearly dependent, then both sides of this equation equal 0.

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