If `A = 2i + 2j-k, B=6i-3j+k,`then `AxxB` will b given by

Options

A. `2i-2j-k`
B. `6i-3j+2k`
C. `i-10j-18k`
D. `i+j+k`

Answer

To find the cross product of the vectors \( A \) and \( B \), we follow these steps:

### Step 1: Write down the vectors
Given:
\[ A = 2i + 2j - k \]
\[ B = 6i - 3j + k \]

### Step 2: Set up the determinant for the cross product
The cross product \( A \times B \) can be computed using the determinant of a matrix formed by the unit vectors \( i, j, k \) and the components of vectors \( A \) and \( B \):
\[ A \times B = \begin{vmatrix} i & j & k \\ 2 & 2 & -1 \\ 6 & -3 & 1 \end{vmatrix} \]

### Step 3: Calculate the determinant
To calculate the determinant, we expand it as follows:
\[ A \times B = i \begin{vmatrix} 2 & -1 \\ -3 & 1 \end{vmatrix} - j \begin{vmatrix} 2 & -1 \\ 6 & 1 \end{vmatrix} + k \begin{vmatrix} 2 & 2 \\ 6 & -3 \end{vmatrix} \]

Calculating each of these 2x2 determinants:
1. For \( i \):
\[ \begin{vmatrix} 2 & -1 \\ -3 & 1 \end{vmatrix} = (2)(1) - (-1)(-3) = 2 - 3 = -1 \]

2. For \( j \):
\[ \begin{vmatrix} 2 & -1 \\ 6 & 1 \end{vmatrix} = (2)(1) - (-1)(6) = 2 + 6 = 8 \]

3. For \( k \):
\[ \begin{vmatrix} 2 & 2 \\ 6 & -3 \end{vmatrix} = (2)(-3) - (2)(6) = -6 - 12 = -18 \]

### Step 4: Combine the results
Putting it all together:
\[ A \times B = -1i - 8j - 18k \]

### Final Result
Thus, the cross product \( A \times B \) is:
\[ A \times B = -i - 8j - 18k \]