Question

Let u and w be complex numbers such that

|u|=5, |w|=3, and |u+w|=6.
Calculate |u+2w| with proof.

Answer 1

To find \(|u + 2w|\), given the complex numbers \(u\) and \(w\) such that \(|u| = 5\), \(|w| = 3\), and \(|u+w| = 6\), we can utilize the properties of magnitudes and the triangle inequality.


Let us denote:
- \(|u| = 5\)
- \(|w| = 3\)
- \(|u + w| = 6\)

### Step 1: Relate magnitudes through the triangle inequality

From the triangle inequality, we have:

\[
|u + w| \leq |u| + |w|
\]
Substituting the given values, we get:

\[
6 \leq 5 + 3
\]
which holds as \(6 = 8\).

### Step 2: Use the reverse triangle inequality

We also know the reverse triangle inequality:

\[
|u + w| \geq ||u| - |w|| = |5 - 3| = 2
\]
This is satisfied since \(6 \geq 2\).

### Step 3: Use the property of magnitudes of complex numbers

Now we know that the examination of \(u\) and \(w\) must result in a certain configuration in the complex plane. Specifically, because \(|u + w| = 6\), we can conclude that \(u\) and \(w\) are arranged in such a way that the vector \(w\) is partially counteracting \(u\).

### Step 4: Use the formula for \(|u + 2w|\)

Next, to find \(|u + 2w|\):

\[
|u + 2w| = |u + w + w| = |(u + w) + w|
\]

Using the triangle inequality again, we obtain:

\[
|u + 2w| \leq |u + w| + |w| = 6 + 3 = 9
\]
and
\[
|u + 2w| \geq ||u + w| - |w|| = |6 - 3| = 3
\]

We thus have:

\[
3 \leq |u + 2w| \leq 9
\]

### Step 5: Reformulate the expression using Pythagorean theorem

Next, we will express \(u\) and \(w\) in terms of coordinates. Let

\[
u = 5e^{i\theta}, \quad w = 3e^{i\phi}
\]

Then,

\[
|u + w| = |5e^{i\theta} + 3e^{i\phi}| = 6
\]

To square the magnitudes, we utilize the formula:

\[
|u + w|^2 = |u|^2 + |w|^2 + 2 \text{Re}(u \overline{w})
\]
Substituting:

\[
36 = 25 + 9 + 2 \text{Re}(u \overline{w})
\]
\[
36 = 34 + 2 \text{Re}(u \overline{w})
\]
\[
2 = 2 \text{Re}(u \overline{w})
\]
\[
\text{Re}(u \overline{w}) = 1
\]

### Step 6: Calculate \(|u + 2w|\)

Now to find \(|u + 2w|\):

\[
|u + 2w|^2 = |u|^2 + |2w|^2 + 2 \text{Re}(u \overline{(2w)}) = 25 + 4 \cdot 9 + 2 \cdot 2 \cdot \text{Re}(u \overline{w})
\]

Calculation gives:

\[
|u + 2w|^2 = 25 + 36 + 4 = 65
\]

Thus,

\[
|u + 2w| = \sqrt{65}
\]

### Final Result

So, the value of \(|u + 2w|\) is:

\[
\boxed{\sqrt{65}}
\]