Exercises for Chapter II Sections 3, 4, and 5

Do NOT use a calculator for any of these problems. Give only exact answers. (See Preface)

1. Convert the following polar forms to rectangular form a + bi.

2. Convert the following rectangular forms to polar form Give angles in radians.

     (a) 1 + i;       (b) 7i;       (c)        (d) -3;      (e) -4 - 4i.

3. Find the rectangular form a + bi of for

4. Do as in Problem 3 for

5. Find the rectangular form for the conjugate of P = -7 + 8i.

6. Find the rectangular form for the conjugate of where a and b are real numbers.

7. Let Find both the polar and the rectangular form of the point symmetric to S with respect to:

    (a) the real axis;       (b) the imaginary axis;      (c) the origin;

    (d) the straight line through O and 1 + i.

8. Do the same as in Problem 7 for instead of S.

9. Let Find in both polar and rectangular form:

    (a) -A;         (b) -B;                  (e) AB;         (f)

10. For A and B of Problem 9, give the rectangular form of A - B and plot O, A, A - B, and -B. What kind of quadrilateral are these four points the vertices of?

11. Let Find the rectangular form of :

12. Let Find in rectangular form.

13. Given that use the Pythagorean Theorem to find r.

14. Given that and b < 0, find b.

15. Explain why for all real a and b and find |11 - 8i|.

16. Use the results of Problem 16 in Chapter 1 to find complex numbers in polar and rectangular form with magnitude 1 and arguments equal to those listed below but converted to radians.

    (a) 105^o;     (b) 15^o;      (c) 225^o;     (d) 135^o;      (e) 195^o.

17. Verify the rules for addition, subtraction, and multiplication of complex numbers given on in Chapter 2 Section 5.

18. Let Show that :

19. Let Find in both polar and rectangular form:

     (a) iA;       (b) -A;       (c) -iA;              (f) A²;       (g) A³.

20. Let Find the polar form (in terms of ) for :

     (a) -20 - 21i;       (b) 21 - 20i;       (c) 20 + 21i;       (d) -21 - 20i;       (e) 20 - 21i.

21. Let Find in both polar and rectangular form.

22. Let Find in both polar and rectangular form.

23. Given that find in rectangular form:

24. Let For each of the following angles find r, a, and b such that

25. Express in a + bi form by first converting to polar form, raising to powers and multiplying in polar form, and finally converting back.

26. Do as in Problem 25 for (5 + 5i)¹¹(-7i)⁸.

27. Let Q = 13, and S = P + Q.

     (a) Explain why O, Q, S, P are the vertices of a rhombus (i.e., a parallelogram with all four sides equal).

     (b) Explain why

     (c) Find the absolute value s of S and then find in a + bi form.

     (d) Let R be as in (c). Explain why R and -R are the 2 square roots of P.

28. Let Q = r, S = P + Q, and s = |S|. Explain why each of the following is true.

     (a) O, Q, S, and P are the vertices of a rhombus.

     (b)

     (c)

     (d) The square roots of P are

29. Let with b > 0. Use Problem 28 to show that the 2 square roots of P are

30. Let with b < 0. Use Problem 28 to show that the 2 square roots of P are

31. Let Show that the square roots of V are

where like signs are used if s > 0 and unlike signs if s < 0.

32. Use Problem 31 to find in a + bi form.

33. Let D = B - A. See figure 13.     (a) Explain why and have the same magnitude and hence |B - A| equals the     distance between A and B.     (b) If A = u + vi and B = x + yi, show that

Figure 13

34. Let A = 5 + 2i, B = 9 + 5i, and C = 5 + 5i.

    (a) Explain why is a right angle.

    (b) Find sides AC and CB of and then use the Pythagorean Theorem to find side AB (i.e., the distance between
    A and B.)

    (c) Find |B - A| and |B| - |A| and tell which equals the distance between A and B.

35. Sketch and identify the locus of all points A in the Argand Plane such that |A - i| = 3. (That is, give the graph in the Argand Plane of the equation |A - i| = 3.)

36. Do as in Problem 35 for each of the following equations.

    (a) |A - 4| + |A - 3i| = 5.

    (b) |A - 4| - |A - 3i| = 0.

37. Let be two complex numbers with the same argument Also let r > 0 and s > 0. Explain why each of the following is true:

38. Perform each of the following divisions in polar form.

    (a)      (b)      (c)

39. Let P and Q be complex numbers with |Q| = s > 0. Show that

40. Use the result of Problem 39 to perform each of the following divisions.

    (a)       (b)       (c)       (d)

41. Show that if then

42. Let A = 6 and B = 4 + 3i.

    (a) Plot the points A and B and complete the

    (b) Find the rectangular form of and

    (c) On the same graph you used for part (a), plot the points A' and B' and complete the How would you
    describe the effect that multiplying by had on the

43. Let A = 5 - i and B = 5 + i. Rotate 30^o counterclockwise about the origin. Find the new coordinates of the vertices and graph the triangle before and after the rotation.

44. Let A = 2 + 5i and B = -1 + 4i. Rotate 135^o clockwise about the origin. Find the new coordinates of the vertices and graph the triangle before and after the rotation.

45. Let A = -1 - 2i and B = 3 - 2i. Rotate 60^o counterclockwise about the origin and at the same time stretch it so that each side is twice the length of the original. Find the new coordinates of the vertices and graph the triangle before and after the rotation.

46. Let A = 6 + 12i and B = -3 + 15i. Rotate 120^o clockwise about the origin and at the same time shrink it so that each side is one third the length of the original. Find the new coordinates of the vertices and graph the triangle before and after the rotation.

47. Let A = 5 - 2i and B = 3 + 2i. Rotate 45^o counterclockwise about the origin and at the same time stretch it so that the area is twice the area of the original. Find the new coordinates of the vertices and graph the triangle before and after the rotation.

48. Let A = 4 + 3i and B = 2 + 5i. Rotate counterclockwise and stretch it until vertex A is at -5 + 15i. Find the new coordinate of B and graph the triangle before and after the rotation.