Question

If \(\overset{̄}{a} = \hat{i}-\hat{k}\), \(\overset{̄}{b} = x\hat{i}+\hat{j}+(1-x)\hat{k}\) and \(\overset{̄}{c} = y\hat{i}+x\hat{j}+(1+x-y)\hat{k}\) then \([\overset{̄}{a} \overset{̄}{b} \overset{̄}{c}]\) depends on

• A. only x
• B. neither x nor y
• C. either x or y
• D. only y

Hint:

Expand the determinant and see which variables remain.

Answer 1

The Correct Option is B

Step 1: Simplify with column operations:
Apply $C_3 \to C_3 + C_1$. The new third column has entries $-1 + 1 = 0$, $(1 - x) + x = 1$ and $(1 + x - y) + y = 1 + x$.

Step 2: Expand:
The determinant is $\begin{vmatrix} 1 & 0 & 0 \\ x & 1 & 1 \\ y & x & 1 + x\end{vmatrix} = 1\cdot[(1)(1 + x) - (1)(x)] = 1$.
A constant, so neither $x$ nor $y$ matters.

Final Answer:
The value is $1$, independent of $x$ and $y$, option (B). \[ \boxed{\text{neither } x \text{ nor } y} \]