Chapter 1
Introduction to N-Dimensional Geometry

1.2 Points, Vectors, and Parallel Lines

1.2.5 Problems

A remark about the exercises is necessary. Certain questions are phrased as statements to avoid the incessant use of "prove that". See Problem 1, for example. Such statements are supposed to be proved. Other questions have a "true-false" or "yes-no" quality. The point of such questions is not to guess, but to justify your answer. Questions marked with are considered to be more challenging. Hints are given for some problems. Of course, a hint may contain statements that must be proved.

1. 1. Let be a nonempty set in . If every three points of are collinear, then is collinear.

   Solution

   
   

   Let and be two distinct points in , then there is a unique line passing through these two points. Now let be an arbitrary point in , from the hypothesis, , and must be on some line , and since and uniquely determine the line , we must have . Therefore, every point in is on the line .

2. 3. Given that the line has the linear equation

   show that the point

   is on the line, and that the vector is parallel to the line.

   Hint. If is on the line and if is also on the line, then must be parallel to the line.

   Solution

   
   

   Substituting the coordinates of this point into the linear equation for , we see that

   so that the given point is on .

   Since not both and are 0, we may assume that , and let

   then both and are on the line , and therefore is parallel to . However,

   so the vector is parallel to .

   Note that this follows immediately from the fact that the vector

   is the normal vector to the line .

3. 5. The centroid of three noncollinear points , and in is defined to be

   Show that this definition of the centroid yields the synthetic definition of the centroid of the triangle with vertices , namely, the point at which the three medians of the triangle intersect. Prove also that the medians do indeed intersect at a common point.

   Solution

   
   

   Given a triangle with vertices , let be the midpoint of the segment and let be the point along the median which is the distance from to .

   We have , and since , then

   If we define and similarly, then we see that

   so that the point lies on each of the three medians. Thus, this is the synthetic definition of the centroid and the medians intersect at a single point.

1.4 Inner Product and Orthogonality

1.4.3 Problems

In the following exercises, assume that "distance" means "Euclidean distance" unless otherwise stated.

1. 1.
   1. a. The unit cube in is the set of points

      Draw the unit cube in , , and .

   2. b. What is the length of the longest line segment that you can place in the unit cube of ?
   3. c. What is the radius of the smallest Euclidean ball that contains the unit cube of ?

   Solution

   
   
   1. a. The unit cubes are sketched below.
   2. b. If and are points in the unit cube in , then the Euclidean distance between and is

      where and for .

      The maximum distance will occur when and for , that is, when and are vertices of the cube that are diagonally opposite. In this case, the maximum distance is

   3. c. The smallest Euclidean ball that contains the unit cube is one that has diameter equal to , the length of the longest line segment in the cube. The ball is

      and has radius .

2. 3. Find the distance between the points and using
   1. a. the metric,
   2. b. the "sup" metric,
   3. c. the Euclidean metric.

   Solution

   
   

   If and , then

   Therefore,

   1. a. With the metric,
   2. b. With the "sup" metric,
   3. c. With the Euclidean metric,
3. 5. Show that a positive homothet of a closed ball is a closed ball.

   Solution

   
   

   Let , and , we will show that

   Let , then where , and therefore

   so that , and

   Conversely, if , then letting , we have

   that is

   so that , and

1.6 Hyperplanes and Linear Functionals

1.6.3 Problems

In the following exercises, unless otherwise stated, assume that the closed unit ball is the closed unit ball in the Euclidean norm.

1. *1. Find a hyperplane in that is tangent to the unit cube at the point . Verify your answer.

   Solution

   
   

   Let be the linear functional represented by , then the hyperplane

   is tangent to the unit cube at .

   To verify this, note that the point is in , since

   Also, for any point in the cube, for , so that and .

2. 3. Find an equation for the hyperplane of
   1. a. Problem 2 (a),
   2. b. Problem 2 (b).

   Solution

   
   
   1. a. The hyperplane through the point is perpendicular to the line through through the points and . Therefore,

      Since is on , we have

      and the equation of the hyperplane is

   2. b. The hyperplane tangent to the unit sphere at is perpendicular to the line through the points and . Therefore,

      Since is on , we have

      so that

      that is, . The equation of the hyperplane is

3. 5. Given the linear functional , find
   1. a. the point on the closed unit ball where is a maximum,
   2. b. the point in the hyperplane that is closest to the origin,
   3. c. the point in the hyperplane that is closest to the origin.

   Solution

   
   
   1. a. If , then from the Cauchy-Schwarz inequality we have

      and so for all .

      Now let , then

      so that , and

      Therefore, attains its maximum on the closed unit ball at .

   2. b. The hyperplane has equation

      and the line through the points and is perpendicular to and has parametric equations

      for . The point on the hyperplane with minimum Euclidean norm; that is, the point closed to the origin, is the point where the line intersects . Therefore,

      so that , and the closest point on to is .

   3. c. Analogous to part (b), the point on the hyperplane

      closest to the origin is the point with .

      Thus, we want

      that is, .

      The point on closest to the origin is .

4. 7. Let be the linear functional on represented by the vector and let be the set
   1. a. Determine which points of are on the same side of .
   2. b. Which point or points of are closest to ?
   3. c. Which points of are on the same side of as the origin?
   4. d. Find the point or points of that are closest to .

   Solution

   
   

   For each , then value of is given by

   and for , we have

   1. a. The points , and , are on one side of ; that is, in the halfspace

      while the points , , and are on the other side; that is, in the halfspace

   2. b. Since , the point of closest to is the point .
   3. c. Only the point is in the halfspace

      while the points , , , and are in the halfspace

   4. d. Since , while for all other points , the point closest to is .
5. 9. Given that is the hyperplane , and given that , find such that is exactly the same as .

   Solution

   
   

   Note that the point if and only if , that is, if and only if .

   This last equation is true if and only if , that is, if and only if .

   Taking , then , so that , and therefore .

6. 11. Let be the line

   where and are distinct points in , and let be a linear functional on such that and . Find

   1. a. the point where intersects the hyperplane ,
   2. b. the scalar such that the hyperplane passes through the midpoint of the line segment joining and .

   Solution

   
   
   1. a. If is on , then

      so that , and the hyperplane intersects the line at the point .

   2. b. We want

      so that .

7. 13. Given that , where the linear functional on is represented by the vector , find
   1. a. a line through that intersects in exactly one point,
   2. b. a line through that misses .

   Solution

   
   
   1. a. The vector is perpendicular to the hyperplane , and Therefore the line through the origin in the direction of intersects in exactly one point.
   2. b. Since , then . We take the vector and note that

      and the vector is perpendicular to . Thus, and are in , so the line through and lies entirely in and so misses .

8. 15. If the hyperplane intersects the straight line in exactly one point, then for every scalar , the hyperplane

   intersects in exactly one point.

   Hint. Conclude that this must happen because of what we know from Problems 12 and 14.

   Solution

   
   

   If the line misses , then lies in a hyperplane parallel to , which is parallel to . This contradicts the fact that intersects in exactly one point. If the line intersects in more than one point, then ...