3.2.E: Problems on Lines and Planes in \(E^{n}\) (Exercises)
Let \(\overline{a}=(-1,2,0,-7), \overline{b}=(0,0,-1,2),\) and \(\overline{c}=(2,4,-3,-3)\) be points in \(E^{4} .\) Find the symmetric normal equations (see Example \((\mathrm{a}) )\) of the lines \(\overline{a b}, \overline{b c},\) and \(\overline{c a} .\) Are any two of the lines perpendicular? Parallel? On the line \(\overline{a b}\), find some points inside \(L(\overline{a}, \overline{b})\) and some outside \(L[\overline{a}, \overline{b}]\). Also, find the symmetric equations of the line through \(\overline{c}\) that is
\[
\text { (i) parallel to } \overline{a b} ; \quad \text { (ii) perpendicular to } \overline{a b}.
\]
With \(\overline{a}\) and \(\overline{b}\) as in Problem \(1,\) find the equations of the two planes that trisect, and are perpendicular to, the line segment \(L[\overline{a}, \overline{b}] .\)
Given a line \(\overline{x}=\overline{a}+t \vec{u}(\vec{u}=\overline{b}-\overline{a} \neq \overrightarrow{0})\) in \(E^{n},\) define \(f : E^{1} \rightarrow E^{n}\) by
\[
f(t)=\overline{a}+t \vec{u} \text { for } t \in E^{1}.
\]
Show that \(L[\overline{a}, \overline{b}]\) is exactly the \(f\) -image of the interval \([0,1]\) in \(E^{1},\) with \(f(0)=a\) and \(f(1)=b,\) while \(f\left[E^{1}\right]\) is the entire line. Also show that \(f\) is one to one.
\(\left[\text { Hint: } t \neq t^{\prime} \text { implies }\left|f(t)-f\left(t^{\prime}\right)\right| \neq 0 . \text { Why? }\right]\)
A map \(f : E^{n} \rightarrow E^{1}\) is called a linear functional iff
\[
\left(\forall \overline{x}, \overline{y} \in E^{n}\right)\left(\forall a, b \in E^{1}\right) \quad f(a \overline{x}+b \overline{y})=a f(\overline{x})+b f(\overline{y}).
\]
Show by induction that \(f\) preserves linear combinations; that is,
\[
f\left(\sum_{k=1}^{m} a_{k} \overline{x}_{k}\right)=\sum_{k=1}^{m} a_{k} f\left(\overline{x}_{k}\right)
\]
for any \(a_{k} \in E^{1}\) and \(\overline{x}_{k} \in E^{n}\).
From Problem 4 prove that a map \(f : E^{n} \rightarrow E^{1}\) is a linear functional iff there is \(\vec{u} \in E^{n}\) such that
\[
(\forall \overline{x} \in E^{n}) f(\overline{x}) = \vec{u} \cdot \overline{x} ("\text{representation theorem}").
\]
[Hint: If \(f\) is a linear functional, write each \(\overline{x} \in E^{n}\) as \(\overline{x}=\sum_{k=1}^{n} x_{k} \overline{e}_{k}(§§1-3, Theorem 2 ). Then
\[
f(\overline{x})=f\left(\sum_{k=1}^{m} x_{k} \overline{e}_{k}\right)=\sum_{k=1}^{n} x_{k} f\left(\overline{e}_{k}\right).
\]
Setting \(u_{k}=f\left(\overline{e}_{k}\right) \in E^{1}\) and \(\vec{u}=\left(u_{1}, \ldots, u_{n}\right),\) obtain \(f(\overline{x})=\vec{u} \cdot \overline{x},\) as required. For the converse, use Theorem 3 in §§1-3.]
Prove that a set \(A \subseteq E^{n}\) is a plane iff there is a linear functional \(f\) (Problem \(4 ),\) not identically zero, and some \(c \in E^{1}\) such that
\[
A=\left\{\overline{x} \in E^{n} | f(\overline{x})=c\right\}.
\]
(This could serve as a definition of planes in \(E^{n} . )\)
[Hint: \(A\) is a plane iff \(A=\{\overline{x} | \vec{u} \cdot \overline{x}=c\} .\) Put \(f(\overline{x})=\vec{u} \cdot \overline{x}\) and use Problem \(5 .\) Show that \(f \neq 0\) iff \(\vec{u} \neq \overrightarrow{0}\) by Problem 10 of §§1-3.]
Prove that the perpendicular distance of a point \(\overline{p}\) to a plane \(\vec{u} \cdot \overline{x}=c\) in \(E^{n}\) is
\[
\rho\left(\overline{p}, \overline{x}_{0}\right)=\frac{|\vec{u} \cdot \overline{p}-c|}{|\vec{u}|}.
\]
\(\left(\overline{x}_{0} \text { is the orthogonal projection of } \overline{p}, \text { i.e., the point on the plane such }\right.\) that \(\overrightarrow{p x_{0}} \| \vec{u} . )\)
[Hint: Put \(\vec{v}=\vec{u} /|\vec{u}| .\) Consider the line \(\overline{x}=\overline{p}+t \vec{v} .\) Find \(t\) for which \(\overline{p}+t \vec{v}\) lies on both the line and plane. Find \(|t| . ]\)
A globe (solid sphere) in \(E^{n},\) with center \(\overline{p}\) and radius \(\varepsilon>0,\) is the set \(\{\overline{x} | \rho(\overline{x}, \overline{p})<\varepsilon\},\) denoted \(G_{\overline{p}}(\varepsilon) .\) Prove that if \(\overline{a}, \overline{b} \in G_{\overline{p}}(\varepsilon),\) then also \(L[\overline{a}, \overline{b}] \subseteq G_{\tilde{p}}(\varepsilon) .\) Disprove it for the sphere \(S_{\overline{p}}(\varepsilon)=\{\overline{x} | \rho(\overline{x}, \overline{p})=\varepsilon\}\). [Hint: Take a line through \(\overline{p} . ]\)