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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!

 Aug 10, 2024
 #2
avatar+1095 
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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.

 Aug 10, 2024

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