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

Options

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

Answer

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 Cross Product
We will use the distributive property of the cross product to expand the left-hand side:

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

### Step 2: Simplify the Terms
Now we simplify each term:
- The cross product of any vector with itself is zero, so \( \vec{a} \times \vec{a} = 0 \) and \( \vec{b} \times \vec{b} = 0 \).
- We also know that \( \vec{b} \times \vec{a} = -(\vec{a} \times \vec{b}) \).

Thus, the expression simplifies to:

\[ 0 + 9 \vec{a} \times \vec{b} - (-\vec{a} \times \vec{b}) + 0 = 9 \vec{a} \times \vec{b} + \vec{a} \times \vec{b} \]

### Step 3: Combine Like Terms
Now we combine the like terms:

\[ 9 \vec{a} \times \vec{b} + \vec{a} \times \vec{b} = 10 \vec{a} \times \vec{b} \]

### Step 4: Set the Expression Equal to k
Now we have:

\[ 10 \vec{a} \times \vec{b} = k \vec{a} \times \vec{b} \]

### Step 5: 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 \( \vec{a} \times \vec{b} \):

\[ k = 10 \]

Thus, the value of \( k \) is \( 10 \).