vectors `veca and vecb` are inclined at an angle ` theta = 60^(@). " If " |veca|=1, |vecb| =2 , " then " [ (veca + 3vecb) xx ( 3 veca -vecb)] ^(2)` is equal to

Options

A. 225
B. 275
C. 325
D. 300

Explanation

To solve the problem step by step, we will follow the same logical structure as presented in the video transcript.

### Step 1: Write the expression
We start with the expression we need to evaluate:
\[ [(\vec{a} + 3\vec{b}) \times (3\vec{a} - \vec{b})]^2 \]

### Step 2: Expand the cross product
Using the distributive property of the cross product, we can expand the expression:
\[ (\vec{a} + 3\vec{b}) \times (3\vec{a} - \vec{b}) = \vec{a} \times 3\vec{a} + \vec{a} \times (-\vec{b}) + 3\vec{b} \times 3\vec{a} + 3\vec{b} \times (-\vec{b}) \]

This simplifies to:
\[ = 3(\vec{a} \times \vec{a}) - \vec{a} \times \vec{b} + 9(\vec{b} \times \vec{a}) - 3(\vec{b} \times \vec{b}) \]

### Step 3: Simplify using properties of cross products
Using the property that the cross product of any vector with itself is zero (\(\vec{a} \times \vec{a} = 0\) and \(\vec{b} \times \vec{b} = 0\)), we have:
\[ = 0 - \vec{a} \times \vec{b} + 9(\vec{b} \times \vec{a}) + 0 \]
Since \(\vec{b} \times \vec{a} = -(\vec{a} \times \vec{b})\), we can rewrite it as:
\[ = -\vec{a} \times \vec{b} - 9(\vec{a} \times \vec{b}) = -10(\vec{a} \times \vec{b}) \]

### Step 4: Square the result
Now we need to square the result:
\[ [-10(\vec{a} \times \vec{b})]^2 = 100(\vec{a} \times \vec{b})^2 \]

### Step 5: Calculate \((\vec{a} \times \vec{b})^2\)
The magnitude of the cross product \(\vec{a} \times \vec{b}\) can be calculated using the formula:
\[ |\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin(\theta) \]
Given that \(|\vec{a}| = 1\), \(|\vec{b}| = 2\), and \(\theta = 60^\circ\), we have:
\[ |\vec{a} \times \vec{b}| = 1 \cdot 2 \cdot \sin(60^\circ) = 2 \cdot \frac{\sqrt{3}}{2} = \sqrt{3} \]
Thus,
\[ (\vec{a} \times \vec{b})^2 = (\sqrt{3})^2 = 3 \]

### Step 6: Substitute back into the squared expression
Now substituting back, we get:
\[ 100(\vec{a} \times \vec{b})^2 = 100 \cdot 3 = 300 \]

### Final Answer
Therefore, the value of the expression is:
\[ \boxed{300} \]