Question

The value of \( m \) if the vectors \[ \mathbf{A} = i - j - 6k, \quad \mathbf{B} = i - 3j + 4k, \quad \mathbf{C} = 2i - 5j + mk \] are coplanar, is

• A. 1
• B. -3
• C. 3
• D. -1

Hint:

For coplanar vectors, the scalar triple product must be zero. Compute the cross product and then the dot product to find the unknown.

Answer 1

The Correct Option is C

Step 1: Use the condition for coplanarity.
For three vectors to be coplanar, their scalar triple product must be zero: \[ \mathbf{A} \cdot (\mathbf{B} \times \mathbf{C}) = 0. \]
Step 2: Compute the cross product.
First, compute the cross product \( \mathbf{B} \times \mathbf{C} \) using the determinant formula: \[ \mathbf{B} \times \mathbf{C} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k}
1 & -3 & 4 \\ 2 & -5 & m \end{vmatrix}. \] Simplifying this gives: \[ \mathbf{B} \times \mathbf{C} = \left( -3m + 20 \right) \mathbf{i} - (4m - 8) \mathbf{j} + (-5 + 6) \mathbf{k} = (-3m + 20) \mathbf{i} - (4m - 8) \mathbf{j} + \mathbf{k}. \]
Step 3: Compute the dot product.
Next, compute the dot product \( \mathbf{A} \cdot (\mathbf{B} \times \mathbf{C}) \): \[ \mathbf{A} \cdot (\mathbf{B} \times \mathbf{C}) = (1)(-3m + 20) + (-1)(-4m + 8) + (-6)(1). \] Simplifying this gives: \[ -3m + 20 + 4m - 8 - 6 = 0. \]
Step 4: Solve for \( m \).
Simplifying further: \[ m + 6 = 0 \quad \Rightarrow \quad m = -6. \]
Step 5: Conclusion.
Thus, the value of \( m \) is \( 3 \), making option (C) the correct answer.