If `A = 2i + 2j-k, B = 6i-3j+2k, ` then `A xx B` will be 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 will follow these steps:

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

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

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

### Step 4: Calculate each of the 2x2 determinants
1. For the \( i \) component:
\[ \begin{vmatrix} 2 & -1 \\ -3 & 2 \end{vmatrix} = (2)(2) - (-1)(-3) = 4 - 3 = 1 \]

2. For the \( j \) component:
\[ \begin{vmatrix} 2 & -1 \\ 6 & 2 \end{vmatrix} = (2)(2) - (-1)(6) = 4 + 6 = 10 \]

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

### Step 5: Combine the components
Now substituting back into our expression for \( A \times B \):
\[ A \times B = 1i - 10j - 18k \]

### Final Result
Thus, the cross product \( A \times B \) is:
\[ A \times B = i - 10j - 18k \]