If z1 = 3 + 4i, z2 = 5 -12i, and z3 = 6 + 8i, find
|z1| , |z2|, |z3|, |z1 + z2| ,| z2 - z3|, and |z1 + z3|.
Using the given values for z1, z2 and z3 we get |z1| = |3 + 4i| = √[32 +42] =5
|z2| = |5 -12i| = √[52 + (-12)2] = 13
z3 = |6 + 8i| = √[62 + 82] = 10
|z1 + z2| = |(3 + 4i ) + (5 -12i)| = |8 - 8i| =
√128 = 8√2
|z2 - z3| = |(5 -12i ) - (6 + 8i )| = |-1- 20i| =
√401
|z1 + z3| = |(3 + 4i) + (6 + 8i)| = |9 +12i| =
√225 = 15
Note that the triangle inequality is satisfied in all the cases.
|z1 + z3| = |z1| + |z3|= 15 (why?)
Example 2.10
Find the following
Solution
Example 2.11
Which one of the points i, -2 + i , and 3 is
farthest from the origin?
Solution
The distance between origin to z = i, -2 + i,
and 3 are
| z | = | i | = 1
| z | = | -2 + i |= √[(-2)2+(1)2] = √5
= | z | = | 3 | = 3
Since 1 < √5 < 3 , the farthest point from the origin is 3
Example 2.12
If z1, z2, and z3 are complex numbers such that |z1| = |z2| =|z3| = |z1 + z2 + z3| = 1, find the value of .
Solution
Since, |z1| = |z1| = |z1| = 1,
Example 2.13
If |z| = 2 show that 3 ≤ |z + 3 + 4i| ≤ 7
Solution
|z + 3 + 4i| ≤ |z| + |3 + 4i| = 2 + 5 = 7
|z + 3 + 4i| ≤ 7 (1)
|z + 3 + 4i| ≥ | |z| - | 3 + 4i| | = |2-5| = 3
|z + 3 + 4i| ≥ 3
(2)
From (1) and (2), we get 3 ≤ |z + 3 + 4i| ≤ 7
Note
To find the lower bound and upper bound use | |z1| - |z2| | ≤ |z1 + z2 | ≤ |z1| + |z2|.
Example 2.14
Show that the points 1, are
the vertices of an equilateral triangle.
Solution
It is enough to prove that the sides of the triangle are equal.
Let z = 1,
The length of the sides of the triangles are
Since the sides are equal, the given points form an equilateral
triangle.
Example 2.15
Let z1 , z2 , and z3 be complex numbers such that |z1| = |z2| =|z3| = r > 0 and z1 + z2 + z3 ≠0 .
Prove that .
Solution
Example 2.16
Show that the equation z2 = has four solutions.
Solution
We have,
It has 3 non-zero solutions. Hence including zero solution, there
are four solutions.
Related Topics
Privacy Policy, Terms and Conditions, DMCA Policy and Compliant
Copyright © 2018-2024 BrainKart.com; All Rights Reserved. Developed by Therithal info, Chennai.