If vectors `2i - j + k, i + 2j - 3k` and `3i + mj + 5k` are coplanar, then m is a root of the equation

Options

A. `x^(2)` + 3x= 4
B. `x^(2)` + 2x= 6
C. `x^(2)` + 3x= 6
D. `x^(2)` + 2x= 4

Answer

To determine the value of \( m \) such that the vectors \( \mathbf{a} = 2\mathbf{i} - \mathbf{j} + \mathbf{k} \), \( \mathbf{b} = \mathbf{i} + 2\mathbf{j} - 3\mathbf{k} \), and \( \mathbf{c} = 3\mathbf{i} + m\mathbf{j} + 5\mathbf{k} \) are coplanar, we can use the scalar triple product condition. The vectors are coplanar if the scalar triple product \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 0 \).

### Step-by-step Solution:

1. **Define the Vectors**:
\[ \mathbf{a} = 2\mathbf{i} - \mathbf{j} + \mathbf{k} \]
\[ \mathbf{b} = \mathbf{i} + 2\mathbf{j} - 3\mathbf{k} \]
\[ \mathbf{c} = 3\mathbf{i} + m\mathbf{j} + 5\mathbf{k} \]

2. **Calculate the Cross Product \( \mathbf{b} \times \mathbf{c} \)**:
\[ \mathbf{b} \times \mathbf{c} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 2 & -3 \\ 3 & m & 5 \end{vmatrix} \]

Expanding the determinant:
\[ = \mathbf{i} \begin{vmatrix} 2 & -3 \\ m & 5 \end{vmatrix} - \mathbf{j} \begin{vmatrix} 1 & -3 \\ 3 & 5 \end{vmatrix} + \mathbf{k} \begin{vmatrix} 1 & 2 \\ 3 & m \end{vmatrix} \]

Calculating the minors:
\[ = \mathbf{i} (2 \cdot 5 - (-3) \cdot m) - \mathbf{j} (1 \cdot 5 - (-3) \cdot 3) + \mathbf{k} (1 \cdot m - 2 \cdot 3) \]
\[ = \mathbf{i} (10 + 3m) - \mathbf{j} (5 + 9) + \mathbf{k} (m - 6) \]
\[ = (10 + 3m)\mathbf{i} - 14\mathbf{j} + (m - 6)\mathbf{k} \]

3. **Calculate the Dot Product \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \)**:
\[ \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = (2\mathbf{i} - \mathbf{j} + \mathbf{k}) \cdot ((10 + 3m)\mathbf{i} - 14\mathbf{j} + (m - 6)\mathbf{k}) \]
\[ = 2(10 + 3m) - 1(-14) + 1(m - 6) \]
\[ = 20 + 6m + 14 + m - 6 \]
\[ = 6m + m + 20 + 14 - 6 \]
\[ = 7m + 28 \]

4. **Set the Scalar Triple Product to Zero**:
\[ 7m + 28 = 0 \]

5. **Solve for \( m \)**:
\[ 7m = -28 \]
\[ m = -4 \]

### Conclusion:
The value of \( m \) such that the vectors are coplanar is \( m = -4 \).