
# 3.2.E: Problems on Lines and Planes in $$E^{n}$$ (Exercises)


Exercise $$\PageIndex{1}$$

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
$\text { (i) parallel to } \overline{a b} ; \quad \text { (ii) perpendicular to } \overline{a b}.$

Exercise $$\PageIndex{2}$$

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 $$L[\overline{a}, \overline{b}] .$$

Exercise $$\PageIndex{3}$$

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.
$$\left[\text { Hint: } t \neq t^{\prime} \text { implies }\left|f(t)-f\left(t^{\prime}\right)\right| \neq 0 . \text { Why? }\right]$$

Exercise $$\PageIndex{4}$$

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 $$\overline{x}_{k} \in E^{n}$$.

Exercise $$\PageIndex{5}$$

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.]

Exercise $$\PageIndex{6}$$

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.]

Exercise $$\PageIndex{7}$$

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 $$|t| . ]$$

Exercise $$\PageIndex{8}$$

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 $$\overline{p} . ]$$