Question

If the vectors \[ \mathbf{a} = 2i - j + k, \quad \mathbf{b} = i + 2j - 3k, \quad \mathbf{c} = 3i + pj + 5k \] are coplanar, find \( p \).

• A. \( 4 \)
• B. \( 14 \)
• C. \( -4 \)
• D. \( 41 \)

Hint:

For coplanar vectors, use determinant expansion and solve for unknowns.

Answer 1

The Correct Option is C

Step 1: Condition for Coplanarity
Vectors are coplanar if: \[ \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 0 \] Expanding determinant: \[ \begin{vmatrix} 2 & -1 & 1
1 & 2 & -3
3 & p & 5 \end{vmatrix} = 0 \] Solving: \[ 2 \begin{vmatrix} 2 & -3
p & 5 \end{vmatrix} - (-1) \begin{vmatrix} 1 & -3
3 & 5 \end{vmatrix} + 1 \begin{vmatrix} 1 & 2
3 & p \end{vmatrix} = 0 \] \[ 2 (10 + 3p) + (5 + 9) + (p - 6) = 0 \] \[ 20 + 6p + 14 + p - 6 = 0 \] \[ 6p + p + 28 = 0 \] \[ 7p = -28 \] \[ p = -4 \] Thus, the correct answer is \( -4 \).