The magnitudes of the X and Y components of `overset(rarr)P` are 7 and 6. The magnitudes of the x-and y-components of `overset(rarr)P+overset(rarr)Q` are 11 and 9, respectively. What is the magnitude of Q?

Explanation

To solve the problem, we need to find the magnitude of vector **Q** given the components of vectors **P** and **P + Q**.

### Step-by-Step Solution:

1. **Identify the components of vector P:**
- The x-component of vector **P** is given as 7.
- The y-component of vector **P** is given as 6.
- Therefore, we can express vector **P** as:
\[ \vec{P} = 7\hat{i} + 6\hat{j} \]

2. **Identify the components of vector P + Q:**
- The x-component of vector **P + Q** is given as 11.
- The y-component of vector **P + Q** is given as 9.
- Therefore, we can express vector **P + Q** as:
\[ \vec{P} + \vec{Q} = 11\hat{i} + 9\hat{j} \]

3. **Find the components of vector Q:**
- To find vector **Q**, we can use the relationship:
\[ \vec{Q} = (\vec{P} + \vec{Q}) - \vec{P} \]
- Substituting the known values:
\[ \vec{Q} = (11\hat{i} + 9\hat{j}) - (7\hat{i} + 6\hat{j}) \]
- Performing the subtraction:
\[ \vec{Q} = (11 - 7)\hat{i} + (9 - 6)\hat{j} = 4\hat{i} + 3\hat{j} \]

4. **Calculate the magnitude of vector Q:**
- The magnitude of vector **Q** can be calculated using the formula:
\[ |\vec{Q}| = \sqrt{(Q_x)^2 + (Q_y)^2} \]
- Substituting the components of **Q**:
\[ |\vec{Q}| = \sqrt{(4)^2 + (3)^2} = \sqrt{16 + 9} = \sqrt{25} \]
- Therefore, the magnitude of vector **Q** is:
\[ |\vec{Q}| = 5 \]

### Final Answer:
The magnitude of vector **Q** is **5**.

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