`|vec(a)|=2, |vec(b)|=3 and vec(a).vec(b)= -8`, then the value of `|vec(a) + 3vec(b)|` is

Options

A. 37
B. `sqrt37`
C. 6
D. 13

Answer

To solve the problem, we need to find the magnitude of the vector expression \(|\vec{a} + 3\vec{b}|\). We are given the following information:

- \(|\vec{a}| = 2\)
- \(|\vec{b}| = 3\)
- \(\vec{a} \cdot \vec{b} = -8\)

### Step-by-Step Solution:

1. **Use the formula for the magnitude of the sum of vectors:**
\[ |\vec{a} + \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + 2(\vec{a} \cdot \vec{b}) \]
In our case, we need to find \(|\vec{a} + 3\vec{b}|\), so we modify the formula:
\[ |\vec{a} + 3\vec{b}|^2 = |\vec{a}|^2 + |3\vec{b}|^2 + 2(\vec{a} \cdot 3\vec{b}) \]

2. **Calculate \(|3\vec{b}|\):**
\[ |3\vec{b}| = 3|\vec{b}| = 3 \times 3 = 9 \]
Therefore, \(|3\vec{b}|^2 = 9^2 = 81\).

3. **Substitute the values into the formula:**
\[ |\vec{a} + 3\vec{b}|^2 = |\vec{a}|^2 + |3\vec{b}|^2 + 2(\vec{a} \cdot 3\vec{b}) \]
\[ = 2^2 + 9^2 + 2 \cdot 3 \cdot (\vec{a} \cdot \vec{b}) \]
\[ = 4 + 81 + 6(\vec{a} \cdot \vec{b}) \]

4. **Substitute \(\vec{a} \cdot \vec{b} = -8\):**
\[ = 4 + 81 + 6(-8) \]
\[ = 4 + 81 - 48 \]
\[ = 85 - 48 = 37 \]

5. **Take the square root to find the magnitude:**
\[ |\vec{a} + 3\vec{b}| = \sqrt{37} \]

### Final Answer:
\[ |\vec{a} + 3\vec{b}| = \sqrt{37} \]