If `(3 vec(a)-vec(b))xx(vec(a) + 3 vec(b)) =k vec(a)xxvec(b)` then what is the value of `k`?

Options

A. `10`
B. `5`
C. `8`
D. `-8`

Explanation

To solve the equation \( (3 \vec{a} - \vec{b}) \times (\vec{a} + 3 \vec{b}) = k \vec{a} \times \vec{b} \), we will follow these steps:

### Step 1: Expand the Left-Hand Side (LHS)
We start with the expression on the left-hand side:
\[ \text{LHS} = (3 \vec{a} - \vec{b}) \times (\vec{a} + 3 \vec{b}) \]
Using the distributive property of the cross product, we can expand this:
\[ \text{LHS} = (3 \vec{a}) \times \vec{a} + (3 \vec{a}) \times (3 \vec{b}) - \vec{b} \times \vec{a} - \vec{b} \times (3 \vec{b}) \]

### Step 2: Simplify Each Term
Now we simplify each term:
1. \( (3 \vec{a}) \times \vec{a} = 0 \) (because the cross product of any vector with itself is zero).
2. \( (3 \vec{a}) \times (3 \vec{b}) = 9 (\vec{a} \times \vec{b}) \).
3. \( -\vec{b} \times \vec{a} = -1 (\vec{a} \times \vec{b}) \) (using the property \( \vec{b} \times \vec{a} = -(\vec{a} \times \vec{b}) \)).
4. \( -\vec{b} \times (3 \vec{b}) = 0 \) (again, the cross product of a vector with itself is zero).

Putting it all together:
\[ \text{LHS} = 0 + 9 (\vec{a} \times \vec{b}) - (\vec{a} \times \vec{b}) + 0 \]
\[ \text{LHS} = 8 (\vec{a} \times \vec{b}) \]

### Step 3: Set LHS Equal to RHS
Now we set the left-hand side equal to the right-hand side:
\[ 8 (\vec{a} \times \vec{b}) = k (\vec{a} \times \vec{b}) \]

### Step 4: Solve for \( k \)
Since \( \vec{a} \times \vec{b} \) is not zero (assuming \( \vec{a} \) and \( \vec{b} \) are not parallel), we can divide both sides by \( \vec{a} \times \vec{b} \):
\[ k = 8 \]

### Final Answer
Thus, the value of \( k \) is \( \boxed{8} \).

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