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 find the magnitude of vector **Q**, we can follow these steps:

### Step 1: Define the vectors
We are given the components of vector **P**:
- \( P_x = 7 \) (X-component)
- \( P_y = 6 \) (Y-component)

Thus, we can express vector **P** as:
\[ \vec{P} = 7\hat{i} + 6\hat{j} \]

### Step 2: Define the resultant vector
We are also given the components of the resultant vector **P + Q**:
- \( (P + Q)_x = 11 \) (X-component)
- \( (P + Q)_y = 9 \) (Y-component)

Thus, we can express the resultant vector as:
\[ \vec{P} + \vec{Q} = 11\hat{i} + 9\hat{j} \]

### Step 3: Set up the equation for vector Q
From the equation of the resultant vector, we can express vector **Q** as:
\[ \vec{Q} = (\vec{P} + \vec{Q}) - \vec{P} \]

Substituting the known values:
\[ \vec{Q} = (11\hat{i} + 9\hat{j}) - (7\hat{i} + 6\hat{j}) \]

### Step 4: Perform the subtraction
Now, we perform the vector subtraction:
\[ \vec{Q} = (11 - 7)\hat{i} + (9 - 6)\hat{j} \]
\[ \vec{Q} = 4\hat{i} + 3\hat{j} \]

### Step 5: 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**:
\[ |\vec{Q}| = \sqrt{4^2 + 3^2} \]
\[ |\vec{Q}| = \sqrt{16 + 9} \]
\[ |\vec{Q}| = \sqrt{25} \]
\[ |\vec{Q}| = 5 \]

### Final Answer
The magnitude of vector **Q** is \( 5 \).

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