3.2.E: Problems on Lines and Planes in (Exercises)

Exercise

Let and be points in Find the symmetric normal equations (see Example of the lines and Are any two of the lines perpendicular? Parallel? On the line , find some points inside and some outside . Also, find the symmetric equations of the line through that is

Exercise

With and as in Problem find the equations of the two planes that trisect, and are perpendicular to, the line segment

Exercise

Given a line in define by

Show that is exactly the -image of the interval in with and while is the entire line. Also show that is one to one.

Exercise

A map is called a linear functional iff

Show by induction that preserves linear combinations; that is,

for any and .

Exercise

From Problem 4 prove that a map is a linear functional iff there is such that

[Hint: If is a linear functional, write each as and obtain as required. For the converse, use Theorem 3 in §§1-3.]

Exercise

Prove that a set is a plane iff there is a linear functional (Problem not identically zero, and some such that

(This could serve as a definition of planes in
[Hint: is a plane iff Put and use Problem Show that iff by Problem 10 of §§1-3.]

Exercise

Prove that the perpendicular distance of a point to a plane in is

that
[Hint: Put Consider the line Find for which lies on both the line and plane. Find

Exercise

A globe (solid sphere) in with center and radius is the set denoted Prove that if then also Disprove it for the sphere . [Hint: Take a line through