If `vec(a)` and `vec( b)` are inclined at an angle of `120^(@)` and `|vec(a)| =1, |vec(b)|=2` , then t`((vec(a) + 3 vec(b)) xx (3 vec(a) - vec(b)))^(2)` is equal to

Options

A. 225
B. 250
C. 275
D. 300

Answer

To solve the problem, we need to calculate the expression \((\vec{a} + 3\vec{b}) \times (3\vec{a} - \vec{b})\) and then square its magnitude. We are given that \(|\vec{a}| = 1\), \(|\vec{b}| = 2\), and the angle between \(\vec{a}\) and \(\vec{b}\) is \(120^\circ\).

### Step-by-Step Solution:

1. **Identify the Magnitudes and Angle**:
- \(|\vec{a}| = 1\)
- \(|\vec{b}| = 2\)
- \(\theta = 120^\circ\)

2. **Calculate the Cross Product**:
We need to expand the expression \((\vec{a} + 3\vec{b}) \times (3\vec{a} - \vec{b})\):
\[ (\vec{a} + 3\vec{b}) \times (3\vec{a} - \vec{b}) = \vec{a} \times 3\vec{a} + \vec{a} \times (-\vec{b}) + 3\vec{b} \times 3\vec{a} + 3\vec{b} \times (-\vec{b}) \]

- The term \(\vec{a} \times 3\vec{a} = 0\) (cross product of a vector with itself is zero).
- The term \(3\vec{b} \times (-\vec{b}) = 0\) (same reason).

Thus, we simplify to:
\[ \vec{a} \times (-\vec{b}) + 9\vec{b} \times \vec{a} = -\vec{a} \times \vec{b} + 9\vec{b} \times \vec{a} \]
Using the property of cross products \(\vec{b} \times \vec{a} = -(\vec{a} \times \vec{b})\):
\[ -\vec{a} \times \vec{b} - 9(\vec{a} \times \vec{b}) = -10(\vec{a} \times \vec{b}) \]

3. **Magnitude of the Cross Product**:
The magnitude of \(\vec{a} \times \vec{b}\) is given by:
\[ |\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin(\theta) \]
Substituting the known values:
\[ |\vec{a} \times \vec{b}| = 1 \cdot 2 \cdot \sin(120^\circ) \]
We know \(\sin(120^\circ) = \frac{\sqrt{3}}{2}\):
\[ |\vec{a} \times \vec{b}| = 2 \cdot \frac{\sqrt{3}}{2} = \sqrt{3} \]

4. **Calculate the Magnitude of the Cross Product**:
Therefore, we have:
\[ | -10(\vec{a} \times \vec{b}) | = 10 |\vec{a} \times \vec{b}| = 10 \cdot \sqrt{3} \]

5. **Square the Magnitude**:
Now, we need to square this magnitude:
\[ | -10(\vec{a} \times \vec{b}) |^2 = (10 \sqrt{3})^2 = 100 \cdot 3 = 300 \]

Thus, the final answer is:
\[ \boxed{300} \]