For what value of m are the vector `2hat(i) - 3hat(j) + 4hat(k), hat(i) + 2hat(j)- hat(k) and m hat(i) - hat(j)+ 2hat(k)` coplanar?

Options

A. 0
B. `(5)/(3)`
C. 1
D. `(8)/(5)`

Answer

To determine the value of \( m \) for which the vectors \( \mathbf{a} = 2\hat{i} - 3\hat{j} + 4\hat{k} \), \( \mathbf{b} = \hat{i} + 2\hat{j} - \hat{k} \), and \( \mathbf{c} = m\hat{i} - \hat{j} + 2\hat{k} \) are coplanar, we can use the condition that the scalar triple product of the vectors must be zero. This can be represented using the determinant of a matrix formed by the vectors.

### Step-by-Step Solution:

1. **Write the vectors in matrix form**:
We need to form a matrix using the coefficients of the vectors:
\[ \begin{vmatrix} 2 & -3 & 4 \\ 1 & 2 & -1 \\ m & -1 & 2 \end{vmatrix} \]

2. **Calculate the determinant**:
To find the determinant, we can expand it using the first row:
\[ \text{Det} = 2 \begin{vmatrix} 2 & -1 \\ -1 & 2 \end{vmatrix} - (-3) \begin{vmatrix} 1 & -1 \\ m & 2 \end{vmatrix} + 4 \begin{vmatrix} 1 & 2 \\ m & -1 \end{vmatrix} \]

3. **Calculate the 2x2 determinants**:
- For the first determinant:
\[ \begin{vmatrix} 2 & -1 \\ -1 & 2 \end{vmatrix} = (2)(2) - (-1)(-1) = 4 - 1 = 3 \]
- For the second determinant:
\[ \begin{vmatrix} 1 & -1 \\ m & 2 \end{vmatrix} = (1)(2) - (-1)(m) = 2 + m \]
- For the third determinant:
\[ \begin{vmatrix} 1 & 2 \\ m & -1 \end{vmatrix} = (1)(-1) - (2)(m) = -1 - 2m \]

4. **Substituting back into the determinant**:
Now substituting these values back into the determinant:
\[ \text{Det} = 2(3) + 3(2 + m) + 4(-1 - 2m) \]
Simplifying this:
\[ = 6 + 6 + 3m - 4 - 8m \]
\[ = 8 - 5m \]

5. **Set the determinant to zero for coplanarity**:
For the vectors to be coplanar, we set the determinant equal to zero:
\[ 8 - 5m = 0 \]

6. **Solve for \( m \)**:
Rearranging gives:
\[ 5m = 8 \implies m = \frac{8}{5} \]

### Final Answer:
The value of \( m \) for which the vectors are coplanar is \( m = \frac{8}{5} \).