64 CHAPTER 5. FUNDAMENTALS

5.7 Exercises1. Verify all the properties 5.3-5.10.

2. Compute the following

(a) 5(1,2,3,−2)+6(2,1,−2,7)

(b) 5(1,2,−2)−6(2,1,−2)

(c) −3(1,0,3,−2)+(2,0,−2,1)

(d) −3(1,−2,−3,−2)−2(2,−1,−2,7)

(e) −(2,2,−3,−2)+2(2,4,−2,7)

3. Find symmetric equations for the line through the points (2,2,4) and (−2,3,1) .

4. Find symmetric equations for the line through the points (1,2,4) and (−2,1,1) .

5. Symmetric equations for a line are given. Find parametric equations of the line.

(a) x+13 = 2y+3

2 = z+7

(b) 2x−13 = 2y+3

6 = z−7

(c) x+13 = 2y+3 = 2z−1

(d) 1−2x3 = 3−2y

2 = z+1

(e) x−13 = 2y−3

5 = z+2

(f) x+13 = 3−y

5 = z+1

6. Parametric equations for a line are given. Find symmetric equations for the line ifpossible. If it is not possible to do it explain why.

(a) x = 1+2t,y = 3− t,z = 5+3t

(b) x = 1+ t,y = 3− t,z = 5−3t

(c) x = 1+2t,y = 3+ t,z = 5+3t

(d) x = 1−2t,y = 1,z = 1+ t

(e) x = 1− t,y = 3+2t,z = 5−3t

(f) x = t,y = 3− t,z = 1+ t

7. The first point given is a point contained in the line. The second point given is adirection vector for the line. Find parametric equations for the line, determined bythis information.

(a) (1,2,1) ,(2,0,3)

(b) (1,0,1) ,(1,1,3)

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

(d) (1,0,−6) ,(−2,−1,3)

(e) (−1,−2,−1) ,(2,1,−1)