Question

If the vectors \( \vec{a} = \hat{i} - 2\hat{j} + \hat{k} \), \( \vec{b} = 2\hat{i} - 5\hat{j} + p\hat{k} \) and \( \vec{c} = 5\hat{i} - 9\hat{j} + 4\hat{k} \) are coplanar, then the value of \( p \) is

• A. \( -3 \)
• B. \( 3 \)
• C. \( \dfrac{1}{3} \)
• D. \( -\dfrac{1}{3} \)

Hint:

For coplanar vectors, always equate the scalar triple product to zero.

Answer 1

The Correct Option is B

Step 1: Use the condition for coplanarity.
Three vectors are coplanar if the scalar triple product is zero: \[ \vec{a} \cdot (\vec{b} \times \vec{c}) = 0 \]

Step 2: Form the determinant.
\[ \begin{vmatrix} 1 & -2 & 1 \\ 2 & -5 & p \\ 5 & -9 & 4 \end{vmatrix} = 0 \]

Step 3: Evaluate the determinant.
\[ 1(-5 \cdot 4 + 9p) + 2(2 \cdot 4 - 5p) + 1(2 \cdot -9 + 5 \cdot 5) = 0 \] \[ (-20 + 9p) + 2(8 - 5p) + (-18 + 25) = 0 \] \[ -20 + 9p + 16 - 10p + 7 = 0 \] \[ 3 - p = 0 \]

Step 4: Final conclusion.
\[ p = \boxed{3} \]