Question

The number of integral values of $p$ for which the vectors $(p + 1)\hat{i} - 3\hat{j} + p\hat{k},\; p\hat{i} + (p + 1)\hat{j} - 3\hat{k}$ and $-3\hat{i} + p\hat{j} + (p + 1)\hat{k}$ are linearly dependent, is ______. 

• A. 0
• B. 1
• C. 2
• D. 3

Hint:

Dependent vectors in 3D $\implies$ Volume of parallelepiped $= 0$.

Answer 1

The Correct Option is C

Step 1: Determinant Condition
Linearly dependent $\implies$ Scalar Triple Product $= 0$.
$\begin{vmatrix} p+1 & -3 & p \\ p & p+1 & -3 \\ -3 & p & p+1 \end{vmatrix} = 0$.
Step 2: Simplify Determinant
Apply $C_1 \to C_1 + C_2 + C_3$:
$(2p-2) \begin{vmatrix} 1 & -3 & p \\ 1 & p+1 & -3 \\ 1 & p & p+1 \end{vmatrix} = 0$.
$p=1$ is one solution.
Step 3: Expansion
Expanding the remaining determinant leads to a quadratic in $p$. Solving this, you get another real value. There are 2 such values.
Final Answer: (C)