5.7. EXERCISES 65

(f) (0,0,0) ,(2,−3,1)

8. 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

(c) x = 7+ t,y = 3+4t,z = 5−3t

(d) x = 2t,y =−3t,z = 3t

(e) x = 2t,y = 3+2t,z = 5+ t

(f) x = t,y = 3+3t,z = 5+ t

9. 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)

(c) (1,1,0) ,(0,1,2)

(d) (0,1,3) ,(0,3,0)

(e) (0,1,0) ,(0,6,2)

(f) (0,1,2) ,(2,0,2)

10. Draw a picture of the points in R2 which are determined by the following orderedpairs.

(a) (1,2)

(b) (−2,−2)

(c) (−2,3)

(d) (2,−5)

11. Does it make sense to write (1,2)+(2,3,1)? Explain.

12. Draw a picture of the points in R3 which are determined by the following orderedtriples.

(a) (1,2,0)

(b) (−2,−2,1)

(c) (−2,3,−2)

13. 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.

14. 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.