If `abs(a)=2, abs(b)=3" and "abs(2a-b)=5," then "abs(2a+b)` equals

Options

A. 17
B. 7
C. 5
D. 1

Answer

To solve the problem, we need to find the value of \( |2\mathbf{a} + \mathbf{b}| \) given that \( |\mathbf{a}| = 2 \), \( |\mathbf{b}| = 3 \), and \( |2\mathbf{a} - \mathbf{b}| = 5 \).

### Step-by-Step Solution:

1. **Use the given information**:
We know:
\[ |\mathbf{a}| = 2 \quad \text{and} \quad |\mathbf{b}| = 3 \]
This means:
\[ |\mathbf{a}|^2 = 4 \quad \text{and} \quad |\mathbf{b}|^2 = 9 \]

2. **Express the magnitude of \( |2\mathbf{a} - \mathbf{b}| \)**:
We have:
\[ |2\mathbf{a} - \mathbf{b}| = 5 \]
Squaring both sides gives:
\[ |2\mathbf{a} - \mathbf{b}|^2 = 25 \]

3. **Expand the squared magnitude**:
Using the formula for the magnitude of a vector:
\[ |2\mathbf{a} - \mathbf{b}|^2 = (2\mathbf{a} - \mathbf{b}) \cdot (2\mathbf{a} - \mathbf{b}) = 4|\mathbf{a}|^2 - 4(\mathbf{a} \cdot \mathbf{b}) + |\mathbf{b}|^2 \]
Substituting the known values:
\[ 25 = 4(4) - 4(\mathbf{a} \cdot \mathbf{b}) + 9 \]
Simplifying this:
\[ 25 = 16 - 4(\mathbf{a} \cdot \mathbf{b}) + 9 \]
\[ 25 = 25 - 4(\mathbf{a} \cdot \mathbf{b}) \]

4. **Solve for \( \mathbf{a} \cdot \mathbf{b} \)**:
Rearranging gives:
\[ 0 = -4(\mathbf{a} \cdot \mathbf{b}) \implies \mathbf{a} \cdot \mathbf{b} = 0 \]
This means that vectors \( \mathbf{a} \) and \( \mathbf{b} \) are orthogonal.

5. **Calculate \( |2\mathbf{a} + \mathbf{b}| \)**:
Now we need to find \( |2\mathbf{a} + \mathbf{b}|^2 \):
\[ |2\mathbf{a} + \mathbf{b}|^2 = (2\mathbf{a} + \mathbf{b}) \cdot (2\mathbf{a} + \mathbf{b}) = 4|\mathbf{a}|^2 + 4(\mathbf{a} \cdot \mathbf{b}) + |\mathbf{b}|^2 \]
Substituting the known values:
\[ |2\mathbf{a} + \mathbf{b}|^2 = 4(4) + 4(0) + 9 \]
\[ |2\mathbf{a} + \mathbf{b}|^2 = 16 + 0 + 9 = 25 \]

6. **Take the square root**:
Thus, we find:
\[ |2\mathbf{a} + \mathbf{b}| = \sqrt{25} = 5 \]

### Final Answer:
\[ |2\mathbf{a} + \mathbf{b}| = 5 \]