+38 Let u and w be complex numbers such that
|u|=5, |w|=3, and |u+w|=6.
Calculate |u+2w| with proof.
Could someone explain to me the steps to do this? I dont really understand this subject.
Thanks!
+1286 Finding |u+2w|
Understanding the Problem
We are given three pieces of information about complex numbers u and w:
|u| = 5
|w| = 3
|u+w| = 6
We need to find the value of |u+2w|.
Solution Approach
We will use the Law of Cosines for complex numbers to solve this problem.
Applying the Law of Cosines
Let θ be the angle between u and w. Then, we have:
|u+w|^2 = |u|^2 + |w|^2 + 2|u||w|cosθ
Substituting the given values:
6^2 = 5^2 + 3^2 + 253*cosθ
36 = 34 + 30cosθ
cosθ = 1/15
Now, consider |u+2w|^2:
|u+2w|^2 = |u|^2 + (2|w|)^2 + 2|u|(2|w|)cosθ
|u+2w|^2 = 5^2 + (23)^2 + 25*(23)(1/15)
|u+2w|^2 = 25 + 36 + 12
|u+2w|^2 = 73
Therefore, |u+2w| = √73.
So, the value of |u+2w| is √73.
