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!

JRFPLSHelp Aug 10, 2024

#2**0 **

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.

learnmgcat Aug 10, 2024