282 CHAPTER 13. ALGEBRA AND GEOMETRY OF Rp

7. Parametric equations for a line are given. Determine a direction vector for this line.

(a) x = 1+2t,y = 3− t,z = 5+3t

(b) x = 1+ t,y = 3+3t,z = 5− t

8. A line contains the given two points. Find parametric equations for this line. Identifythe direction vector.

(a) (0,1,0) ,(2,1,2)

(b) (0,1,1) ,(2,5,0)

9. Describe in words how to get to the points described by the ordered pairs.

(a) (1,2)

(b) (−2,−2)

10. Does it make sense to write(

1 2)+(

2 3 1)? Explain.

11. Describe in words how to get to the point in R3 denoted by the ordered triples.

(a) (1,2,0)

(b) (−2,−2,1)

(c) (−2,3,−2)

12. You are given two points in R3,(4,5,−4) and (2,3,0) . Show the distance from thepoint (3,4,−2) to the first of these points is the same as the distance from this pointto the second of the original pair of points. Note that 3 = 4+2

2 ,4 = 5+32 . Obtain a

theorem which will be valid for general pairs of points (x,y,z) and (x1,y1,z1) andprove your theorem using the distance formula.

13. A sphere is the set of all points which are at a given distance from a single givenpoint. Find an equation for the sphere which is the set of all points that are at adistance of 4 from the point (1,2,3) in R3.

14. A parabola is the set of all points (x,y) in the plane such that the distance from thepoint (x,y) to a given point (x0,y0) equals the distance from (x,y) to a given line.The point (x0,y0) is called the focus and the line is called the directrix. Find theequation of the parabola which results from the line y = l and (x0,y0) a given focuswith y0 < l. Repeat for y0 > l.

15. Suppose the distance between (x,y) and (x′,y′) were defined to equal the larger ofthe two numbers |x− x′| and |y− y′| . Draw a picture of the sphere centered at thepoint (0,0) if this notion of distance is used.

16. Repeat the same problem except this time let the distance between the two points be|x− x′|+ |y− y′| .

17. If (x1,y1,z1) and (x2,y2,z2) are two points such that |(xi,yi,zi)|= 1 for i = 1,2, showthat in terms of the usual distance,

∣∣( x1+x22 , y1+y2

2 , z1+z22

)∣∣< 1. What would happen ifyou used the way of measuring distance given in Problem 15 (|(x,y,z)|= maximumof |z| , |x| , |y| .)?