Applications and Problems

We shall present a few applications of the technique of vectors to geometry, followed by some problems of more than routine interest. This list should be supplemented by a number of the problems to be found in any of the standard texts on analytic geometry of two and three dimensions.

Intersection of Two Planes in 3-Space

Consider the problem of finding the intersection of two planes in 3-space. In general, this will be a line; we can derive its equation from those of the planes in several ways:

We could solve simultaneously the equations of the planes.
We could find two points which are common to both planes, for instance by choosing two different fixed values for their third coordinates. We then obtain in each case a system of two linear equations in two unknowns (the first and second coordinates), which we solve. Then we find the connecting line.
We could find one point common to both planes, and then find a vector parallel to the line of intersection. The latter is accomplished by observing that any such vector must be perpendicular to each of the known vectors perpendicular to the individual planes. From the exercises on vector products, we see that we can take as our vector the vector product . The formula gives the line.

Distance from a Point to a Line in n-Space

Let be a line in n-space, a point not on the line; let us find the foot of the perpendicular from to this line.

If is this point, then for some . Moreover, the segment connecting with , or the vector , is to be perpendicular to the line , therefore to the vector . Now , giving

This is a linear equation, which we solve for . The desired point is then found by substituting this value for in . The distance between and the line is simply .

Distance Between Two Lines

Now let and be two non-intersecting lines. Let and be the endpoints of their common perpendicular. Then and . These give us two linear equations which we solve for and . The resulting points and determine the common perpendicular. The distance between the two lines is .

Angle Between Two Planes

The angle between two planes is found by using the angle between two vectors, i.e., by reading off two vectors perpendicular to the planes from their equations and finding the angle between these vectors. Since two vectors having opposite directions determine the same plane, there are two possible answers, consisting of supplementary angles. We agree to take the angle between and .

PROBLEMS

1. If a is a number, define its absolute value |a| by . Show that , where is a vector.

2. If and are vectors, show that . Use this to prove the triangle inequality: When can this be an equality?

3. In n-space, let . Show the following: a) If are numbers such that , then . b) If is any vector in n-space, there are numbers such that . How many such sets of numbers are there for a given ? c) and if , for all 1 and . d) If is any vector in n-space and , then . e) If is any vector in n-space, and if , then and each .

4. Show by vectors that the diagonals of a rhombus are perpendicular.

5. Show that the medians of the triangle with vertices , meet in . (Observe that the midpoint of the side is .

6. What is the plane through such that any vector perpendicular to the plane forms equal angles with the vectors and ?

7. Show that the planes and are parallel. Find the distance between them.

8. Let and be non-zero vectors in 3-space. Prove that and are parallel if and only if .

9. Prove the law of sines using vectors in 3-space.

10. What is the set of points in 3-space such that the vector from to is perpendicular to the vector from to ? Show that this is the equation of a sphere with center and radius .

11. Let and be vectors, different from 0 . Show that there exist vectors and such that

is parallel to ,
is perpendicular to ,
, and
Show that and are unique.

12. Let and be the points and ; let be the vector ; let be the line through parallel to ; and notice that does not lie on .

Given a point on , compute the distance between and and show that it is .
Show that the distance between and is precisely for just one , being actually for all other points . Compute the coordinates of this .
Show that the vector is perpendicular to the vector A.

13. Keeping the results of (2) in mind, consider (in n-space) a point and a vector whose length is ; let be the line through parallel to ; and let be a point not on .

Show that there is just one point on such that the vector is perpendicular to and compute its coordinates.
Given a point on , compute the distance from to and show that it is . Show that the distance from to is precisely if and only if .

14. Compute (in 3-space) the equation of the plane which passes through the points , and .

Compute the distance between the point and a point in this plane.
Compute the minimum distance and call it .
Show that there is just one point in the plane such that the distance . Compute the coordinates of .
Show that the vector is perpendicular to the plane.

Given three vectors , and (in n-space) show that the relation is not, in general, true.

A cube in n-dimensional space is a set of points ( , ) satisfying for some set of numbers and some . Then the vertices of the cube are the points of the form , where each is either or . An edge of the cube is a line segment joining two vertices and , where for all except one of the ’s.

How many vertices and edges has a cube in 2-space? in 3-space? in 4-space? in n-space? How would you define a 2-dimensional face of an n-dimensional cube ( )? a k-dimensional face ( )? How many of these are there?

Intersection of Two Planes in 3-Space

Distance from a Point to a Line in n-Space

Distance Between Two Lines

Angle Between Two Planes

PROBLEMS

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