If the vectors `-3hat(i)+4hat(j)-2hat(k),hat(i)+2hat(k),hat(i)-phat(j)` are coplanar, then the value of p is

Options

A. `-2`
B. 1
C. `-1`
D. 2

Answer

To determine the value of \( p \) for which the vectors \( \mathbf{a} = -3\hat{i} + 4\hat{j} - 2\hat{k} \), \( \mathbf{b} = \hat{i} + 2\hat{k} \), and \( \mathbf{c} = \hat{i} - p\hat{j} \) are coplanar, we can use the concept of the scalar triple product. The vectors are coplanar if their scalar triple product is zero.

### Step-by-step Solution:

1. **Write the vectors in component form**:
\[ \mathbf{a} = \begin{pmatrix} -3 \\ 4 \\ -2 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}, \quad \mathbf{c} = \begin{pmatrix} 1 \\ -p \\ 0 \end{pmatrix} \]

2. **Set up the determinant for the scalar triple product**:
The scalar triple product can be expressed as the determinant of a matrix formed by the vectors:
\[ \text{Det} = \begin{vmatrix} -3 & 4 & -2 \\ 1 & 0 & 2 \\ 1 & -p & 0 \end{vmatrix} \]

3. **Calculate the determinant**:
We can calculate this determinant using the formula for a \( 3 \times 3 \) matrix:
\[ \text{Det} = -3 \begin{vmatrix} 0 & 2 \\ -p & 0 \end{vmatrix} - 4 \begin{vmatrix} 1 & 2 \\ 1 & 0 \end{vmatrix} - 2 \begin{vmatrix} 1 & 0 \\ 1 & -p \end{vmatrix} \]

Now, we compute each of these \( 2 \times 2 \) determinants:
- \( \begin{vmatrix} 0 & 2 \\ -p & 0 \end{vmatrix} = 0 \cdot 0 - 2 \cdot (-p) = 2p \)
- \( \begin{vmatrix} 1 & 2 \\ 1 & 0 \end{vmatrix} = 1 \cdot 0 - 2 \cdot 1 = -2 \)
- \( \begin{vmatrix} 1 & 0 \\ 1 & -p \end{vmatrix} = 1 \cdot (-p) - 0 \cdot 1 = -p \)

Substituting these back into the determinant:
\[ \text{Det} = -3(2p) - 4(-2) - 2(-p) = -6p + 8 + 2p \]

Simplifying this gives:
\[ \text{Det} = -4p + 8 \]

4. **Set the determinant to zero for coplanarity**:
For the vectors to be coplanar, we set the determinant equal to zero:
\[ -4p + 8 = 0 \]

5. **Solve for \( p \)**:
Rearranging the equation gives:
\[ -4p = -8 \implies p = 2 \]

### Conclusion:
The value of \( p \) for which the vectors are coplanar is \( p = 2 \).