Question

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

Short answer

Therefore,

$$\begin{gathered} a.\hspace{0.33em}V({\overrightarrow{v}}_1,{\overrightarrow{v}}_2,{\overrightarrow{v}}_3,{\overrightarrow{v}}_1\times {\overrightarrow{v}}_2\times {\overrightarrow{v}}_3)=V({\overrightarrow{v}}_1,{\overrightarrow{v}}_2,{\overrightarrow{v}}_3)\Vert ({\overrightarrow{v}}_1\times {\overrightarrow{v}}_2\times {\overrightarrow{v}}_3)\Vert . \\ b.\hspace{0.33em}V({\overrightarrow{v}}_1,{\overrightarrow{v}}_2,{\overrightarrow{v}}_3,{\overrightarrow{v}}_1\times {\overrightarrow{v}}_2\times {\overrightarrow{v}}_3)=\Vert {\overrightarrow{v}}_1\times {\overrightarrow{v}}_2\times {\overrightarrow{v}}_3{\Vert }^2. \\ c.\hspace{0.33em}V({\overrightarrow{v}}_1,\overrightarrow{v_2}\overrightarrow{v_3}=\Vert {\overrightarrow{v}}_1\times {\overrightarrow{v}}_2\times {\overrightarrow{v}}_3\Vert . \end{gathered}$$

Step-by-step solution

To find the relationship. 

a) To find,

$V\left(\overrightarrow{v_1},\overrightarrow{v_2},\overrightarrow{v_3}\right)=||\overrightarrow{v_1}||||\stackrel{\to \perp }{v_2}||||\stackrel{\to \perp }{v_3}||$

On the other hand,

$$\begin{gathered} V\left(\overrightarrow{v_1},\overrightarrow{v_2},\overrightarrow{v_3},\overrightarrow{v_1}\times \overrightarrow{v_2}\times \overrightarrow{v_3}\right)=||\overrightarrow{v_1}||||\stackrel{\to \perp }{v_2}||||\stackrel{\to \perp }{v_3}||||{\left(\overrightarrow{v_1}\times \overrightarrow{v_2}\times \overrightarrow{v_3}\right)}^{\perp }|| \\ V\left(\overrightarrow{v_1},\overrightarrow{v_2},\overrightarrow{v_3},\overrightarrow{v_1}\times \overrightarrow{v_2}\times \overrightarrow{v_3}\right)=V\left(\overrightarrow{v_1}\times \overrightarrow{v_2}\times \overrightarrow{v_3}\right)||{\left(\overrightarrow{v_1}\times \overrightarrow{v_2}\times \overrightarrow{v_3}\right)}^{\perp }|| \\ V\left(\overrightarrow{v_1},\overrightarrow{v_2},\overrightarrow{v_3},\overrightarrow{v_1}\times \overrightarrow{v_2}\times \overrightarrow{v_3}\right)=V\left(\overrightarrow{v_1},\overrightarrow{v_2},\overrightarrow{v_3}\right)||\left(\overrightarrow{v_1}\times \overrightarrow{v_2}\times \overrightarrow{v_3}\right)|| \end{gathered}$$

To express the terms. 

b) To express the terms,

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

When the v→i are linearly dependent.

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

$$\begin{gathered} V\left(\overrightarrow{v_1},\overrightarrow{v_2},\overrightarrow{v_3}\right)=\frac{V\left(\overrightarrow{v_1},\overrightarrow{v_2},\overrightarrow{v_3},\overrightarrow{v_1}\times \overrightarrow{v_2}\times \overrightarrow{v_3}\right)}{||\left(\overrightarrow{v_1}\times \overrightarrow{v_2}\times \overrightarrow{v_3}\right)||} \\ V\left(\overrightarrow{v_1},\overrightarrow{v_2},\overrightarrow{v_3},\overrightarrow{v_1}\right)=\frac{{||\left(\overrightarrow{v_1}\times \overrightarrow{v_2}\times \overrightarrow{v_3}\right)||}^2}{||\left(\overrightarrow{v_1}\times \overrightarrow{v_2}\times \overrightarrow{v_3}\right)||} \\ V\left(\overrightarrow{v_1},\overrightarrow{v_2},\overrightarrow{v_3}\right)=||\left(\overrightarrow{v_1}\times \overrightarrow{v_2}\times \overrightarrow{v_3}\right)|| \end{gathered}$$

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