Question

If the vectors \( i + j + k \), \( i - j + k \) and \( 2i + 3j + mk \) are coplanar, then \( m = \)

• A. 3
• B. -2
• C. 2
• D. -3

Hint:

For vectors to be coplanar, their scalar triple product must be zero. Always check the scalar triple product when given a set of vectors.

Answer 1

The Correct Option is C

Step 1: Using the condition for coplanarity.
Three vectors are coplanar if and only if their scalar triple product is zero. The scalar triple product of vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) is given by: \[ \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 0 \]
Step 2: Substituting the vectors.
Let the vectors be: \[ \mathbf{a} = (1, 1, 1), \quad \mathbf{b} = (1, -1, 1), \quad \mathbf{c} = (2, 3, m) \] We compute the cross product \( \mathbf{b} \times \mathbf{c} \) and then take the dot product with \( \mathbf{a} \). After solving, we find that \( m = 2 \).

Step 3: Conclusion.
Thus, \( m = 2 \), which makes option (C) the correct answer.