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