Question

The cross product in${\mathit{R}}^{\mathbf{n}}$. Consider the vectors ${\overrightarrow{\mathbf{v}}}_{\mathbf{2}}$, ${\overrightarrow{\mathbf{v}}}_{\mathbf{3}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\overrightarrow{\mathbf{v}}}_{\mathbf{n}}$ in role="math" localid="1660390620385" ${\mathit{R}}^{\mathbf{n}}$ . The transformation $\mathit{T}\mathbf{\left(}\overrightarrow{\mathbf{x}}\mathbf{\right)}\mathbf{=}\mathit{d}\mathit{e}\mathit{t}\left[\begin{array}{ccccc}1& |& |& & |\\ \overrightarrow{x}& {\overrightarrow{v}}_2& {\overrightarrow{v}}_3& ...& {\overrightarrow{v}}_n\\ |& |& |& & |\end{array}\right]$ is linear. Therefore, there exists a unique vector role="math" localid="1660390632609" $\overrightarrow{\mathbf{u}}$ in role="math" localid="1660390628687" ${\mathit{R}}^{\mathbf{n}}$such that$\mathit{T}\mathbf{\left(}\overrightarrow{\mathbf{x}}\mathbf{\right)}\mathbf{=}\overrightarrow{\mathbf{x}}\mathbf{.}\overrightarrow{\mathbf{u}}$for all $\overrightarrow{\mathbf{x}}$ in ${\mathit{R}}^{\mathbf{n}}$. Compare this with Exercise $\mathbf{2}\mathbf{.}\mathbf{1}\mathbf{.}\mathbf{43}\mathit{c}$. This vector role="math" localid="1660390639527" $\overrightarrow{\mathbf{u}}$ is called the cross product of ${\overrightarrow{\mathbf{v}}}_{\mathbf{2}}$,${\overrightarrow{\mathbf{v}}}_{\mathbf{3}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\overrightarrow{\mathbf{v}}}_{\mathbf{n}}$ , written as$\overrightarrow{\mathbf{u}}\mathbf{=}{\overrightarrow{\mathbf{v}}}_{\mathbf{2}}\mathbf{\times }{\overrightarrow{\mathbf{v}}}_{\mathbf{3}}\mathbf{\times }\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{\times }{\overrightarrow{\mathbf{v}}}_{\mathbf{n}}\mathbf{.}$In other words, the cross product is defined by the fact that$\overrightarrow{\mathbf{x}}\mathbf{\times }\left({\overrightarrow{v}}_2\times {\overrightarrow{v}}_3\times ...\times {\overrightarrow{v}}_n\right)$$\mathbf{=}\mathit{d}\mathit{e}\mathit{t}\mathbf{\left[}\begin{array}{ccccc}\mathbf{1}& \mathbf{|}& \mathbf{|}& & \mathbf{|}\\ \overrightarrow{\mathbf{x}}& {\overrightarrow{\mathbf{v}}}_{\mathbf{2}}& {\overrightarrow{\mathbf{v}}}_{\mathbf{3}}& \mathbf{.}\mathbf{.}\mathbf{.}& {\overrightarrow{\mathbf{v}}}_{\mathbf{n}}\\ \mathbf{|}& \mathbf{|}& \mathbf{|}& & \mathbf{|}\end{array}\mathbf{\right]}$ for all role="math" localid="1660390666262" $\overrightarrow{\mathbf{x}}$ in role="math" localid="1660390669928" ${\mathit{R}}^{\mathbf{n}}$. Note that the cross product in role="math" localid="1660390676371" ${\mathit{R}}^{\mathbf{n}}$ is defined for role="math" localid="1660390680958" $\mathit{n}\mathbf{-}\mathbf{1}$ vectors only. (For example, you cannot form the cross product of just two vectors in role="math" localid="1660390685393" ${\mathit{R}}^{\mathbf{4}}$.) Since the$\mathit{i}$th component of a vector role="math" localid="1660390689594" $\overrightarrow{\mathbf{w}}$ is ${\overrightarrow{\mathbf{e}}}_{\mathbf{i}}\mathbf{.}\overrightarrow{\mathbf{w}}$, we can find the cross product by components as follows: role="math" localid="1660390696637" $\mathit{i}$ th component of ${\overrightarrow{\mathbf{v}}}_{\mathbf{2}}\mathbf{\times }{\overrightarrow{\mathbf{v}}}_{\mathbf{3}}\mathbf{\times }\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{\times }{\overrightarrow{\mathbf{v}}}_{\mathbf{n}}$.

a. When is ${\overrightarrow{\mathbf{v}}}_{\mathbf{2}}\mathbf{\times }{\overrightarrow{\mathbf{v}}}_{\mathbf{3}}\mathbf{\times }\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{\times }{\overrightarrow{\mathbf{v}}}_{\mathbf{n}}\mathbf{=}\overrightarrow{\mathbf{0}}$? Give your answer in terms of linear independence.

b. Find ${\overrightarrow{\mathbf{e}}}_{\mathbf{2}}\mathbf{\times }{\overrightarrow{\mathbf{e}}}_{\mathbf{3}}\mathbf{\times }\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{\times }{\overrightarrow{\mathbf{e}}}_{\mathbf{n}}$.

c. Show that${\overrightarrow{\mathbf{v}}}_{\mathbf{2}}\mathbf{\times }{\overrightarrow{\mathbf{v}}}_{\mathbf{3}}\mathbf{\times }\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{\times }{\overrightarrow{\mathbf{v}}}_{\mathbf{n}}$is orthogonal to all the vectors ${\overrightarrow{\mathbf{v}}}_{\mathbf{i}}$, for $\mathit{i}\mathbf{=}\mathbf{2}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}\mathit{n}$.

d. What is the relationship between ${\overrightarrow{\mathbf{v}}}_{\mathbf{2}}\mathbf{\times }{\overrightarrow{\mathbf{v}}}_{\mathbf{3}}\mathbf{\times }\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{\times }{\overrightarrow{\mathbf{v}}}_{\mathbf{n}}$ and ${\overrightarrow{\mathbf{v}}}_{\mathbf{3}}\mathbf{\times }{\overrightarrow{\mathbf{v}}}_{\mathbf{2}}\mathbf{\times }\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{\times }{\overrightarrow{\mathbf{v}}}_{\mathbf{n}}$? (We swap the first two factors.)

e. Express det$\left[{\overrightarrow{v}}_2\times {\overrightarrow{v}}_3\times ...\times {\overrightarrow{v}}_n{\overrightarrow{v}}_2{\overrightarrow{v}}_3...{\overrightarrow{v}}_n\right]$ in terms of $||{\overrightarrow{v}}_2\times {\overrightarrow{v}}_3\times ...\times {\overrightarrow{v}}_n||$.
f. How do we know that the cross product of two vectors in ${\mathit{R}}^{\mathbf{3}}$, as defined here, is the same as the standard cross product in ${\mathit{R}}^{\mathbf{3}}$? See Definition A.9 of the Appendix.

Short answer

1. The equation $\overrightarrow{v_2}\times \overrightarrow{v_3}\times \dots \times \overrightarrow{v_n}=0$applies if and only if the given determinant is 0 for any given $\overrightarrow{x}$, which is if and only if ${\overrightarrow{v}}_2,{\overrightarrow{v}}_3,\dots ,\overrightarrow{v_n}$are linearly dependent.
2. $\overrightarrow{u}=\overrightarrow{e_1}$
3. It is proved that that ${\overrightarrow{v}}_2\times {\overrightarrow{v}}_3\times ...\times {\overrightarrow{v}}_n$is orthogonal to all the vectors ${\overrightarrow{v}}_i$, for $i=2,...,n$.
4. ${\overrightarrow{v}}_3\times {\overrightarrow{v}}_2\times \dots \times \overrightarrow{v_n}=-\overrightarrow{v_2}\times {\overrightarrow{v}}_3\times \dots \times {\overrightarrow{v}}_n$
5. $\det \left[\overrightarrow{v_2}\times \overrightarrow{v_3}\times \dots \times {\overrightarrow{v}}_n{\overrightarrow{v}}_2{\overrightarrow{v}}_3\dots \overrightarrow{v_n}\right]={||\overrightarrow{v_2}\times \overrightarrow{v_3}\times \dots \times \overrightarrow{v_n}||}^2$
6. $\overrightarrow{v}\times \overrightarrow{w}=\left(v_2{w}_3-w_2{v}_3,v_3{w}_1-v_1{w}_3,v_1{w}_2-v_2{w}_1\right),$
   

Step-by-step solution

Step by Step Solution: Step 1: Give your answer in terms of linear independence

a. The equation $\overrightarrow{v_2}\times \overrightarrow{v_3}\times \dots \times \overrightarrow{v_n}=0$applies if and only if the given determinant is 0 for any given $\overrightarrow{x}$, which is if and only if${\overrightarrow{v}}_2,{\overrightarrow{v}}_3,\dots ,\overrightarrow{v_n}$ are linearly dependent.

To find e→2×e→3×...×e→n

b. To find$\overrightarrow{e_2}\times \overrightarrow{e_3}\times \dots \times {\overrightarrow{e}}_n$, we have to find the vector$\overrightarrow{u}\in {\mathrm{ℝ}}^n$such that

role="math" localid="1660389198561" $\det \left[\begin{array}{ccccc}\overrightarrow{x}& \overrightarrow{e_2}& \overrightarrow{e_3}& ...& \overrightarrow{e_n}\end{array}\right]=\overrightarrow{x}·\overrightarrow{u},\forall \overrightarrow{x}\in {\mathrm{ℝ}}^n$

This gives us $\overrightarrow{x}·\overrightarrow{u}=x_1$

So,

$\overrightarrow{u}=\left[\begin{array}{c}1\\ 0\\ ⋮\\ 0\end{array}\right]$

$\overrightarrow{u}=\overrightarrow{e_1}$

To show that v→2×v→3×...×v→n.

c. For $i=2,\dots ,n$, if $\overrightarrow{x}={\overrightarrow{v}}_i$, then the first and the $i$-th column of the given matrix are the same, which means the determinant is 0 , giving us

${\overrightarrow{v}}_i·\left(\overrightarrow{v_2}\times {\overrightarrow{v}}_3\times \dots {\overrightarrow{v}}_n\right)=0,\forall i=2,\dots ,n$

To find the relationship.

d. We compute,

$\det \left[\begin{array}{ccccc}\overrightarrow{x}& \overrightarrow{v_2}& \overrightarrow{v_3}& ...& {\overrightarrow{v}}_n\end{array}\right]=-\det \left[\begin{array}{ccccc}\overrightarrow{x}& \overrightarrow{v_3}& \overrightarrow{v_2}& ...& {\overrightarrow{v}}_n\end{array}\right]$

which gives us,

$\overrightarrow{x}·\left(\overrightarrow{v_2}\times \overrightarrow{v_3}\times \dots \times {\overrightarrow{v}}_n\right)=-\overrightarrow{x}·\left(\overrightarrow{v_3}\times \overrightarrow{v_2}\times \dots \times {\overrightarrow{v}}_n\right)$

Thus,

${\overrightarrow{v}}_3\times {\overrightarrow{v}}_2\times \dots \times \overrightarrow{v_n}=-\overrightarrow{v_2}\times {\overrightarrow{v}}_3\times \dots \times {\overrightarrow{v}}_n$

To express v→2×v→3×...×v→nv→2v→3...v→n

e. To express we get,

$$\begin{gathered} \det \left[\begin{array}{cc}\overrightarrow{v_2}\times \overrightarrow{v_3}\times \dots \times {\overrightarrow{v}}_n& {\overrightarrow{v}}_2{\overrightarrow{v}}_3\dots \overrightarrow{v_n}\end{array}\right]=\left(\overrightarrow{v_2}\times \overrightarrow{v_3}\times \dots \times \overrightarrow{v_n}\right)·\left(\overrightarrow{v_2}\times \overrightarrow{v_3}\times \dots \times \overrightarrow{v_n}\right) \\ \det \left[\begin{array}{cc}\overrightarrow{v_2}\times \overrightarrow{v_3}\times \dots \times {\overrightarrow{v}}_n& {\overrightarrow{v}}_2{\overrightarrow{v}}_3\dots \overrightarrow{v_n}\end{array}\right]={\left|\left|\overrightarrow{v_2}\times \overrightarrow{v_3}\times \dots \times \overrightarrow{v_n}\right|\right|}^2 \end{gathered}$$

Step 6: How do calculate the cross product of two vectors

f) If we were to find the cross product of two vectors $\overrightarrow{v},\overrightarrow{w}\in {\mathrm{ℝ}}^3$, as defined here, we would have to solve:

$$\begin{gathered} \left|\begin{array}{ccc}{x}_1& v_1& w_1\\ x_2& v_2& w_2\\ x_3& v_3& w_3\end{array}\right|=\left(x_1,x_2,x_3\right)·\left(u_1,u_2,u_3\right),\forall \left(x_1,x_2,x_3\right)\in {\mathrm{ℝ}}^3 \\ \left(v_2{w}_3-v_3{w}_2\right)x_1-\left(v_1{w}_3-v_3{w}_1\right)x_2+\left(v_1{w}_2-v_2{w}_1\right)x_3=u_1{x}_1+u_2{x}_2+u_3{x}_3 \\ {u}_1=v_2{w}_3-v_3{w}_2,u_2=v_3{w}_1-v_1{w}_3,u_3=v_1{w}_2-v_2{w}_1 \end{gathered}$$

So, $\overrightarrow{v}\times \overrightarrow{w}=\left(v_2{w}_3-v_3{w}_2,v_3{w}_1-v_1{w}_3,v_1{w}_2-v_2{w}_1\right).$

However, if we were to calculate their cross product by the standard definition, we would compute:

$$\begin{gathered} \overrightarrow{v}\times \overrightarrow{w}=\left|\begin{array}{ccc}\overrightarrow{e_1}& \overrightarrow{e_2}& \overrightarrow{e_3}\\ v_1& v_2& v_3\\ w_1& w_2& w_3\end{array}\right| \\ \overrightarrow{v}\times \overrightarrow{w}=\left(v_2{w}_3-w_2{v}_3,v_3{w}_1-v_1{w}_3,v_1{w}_2-v_2{w}_1\right), \end{gathered}$$

This is the same.