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: Understanding the Concept:
The scalar triple product \([\bar a\ \bar b\ \bar c]\) equals the determinant of the components of the three vectors.

Step 2: Key Formula or Approach:
\(\begin{vmatrix} 1 & 0 & -1 \\ x & 1 & 1 - x \\ y & x & 1 + x - y\end{vmatrix}\).

Step 3: Detailed Explanation:
Expand along the first row: \(1\cdot[1(1 + x - y) - x(1 - x)] - 0 + (-1)[x\cdot x - 1\cdot y]\).
First bracket: \(1 + x - y - x + x^2 = 1 - y + x^2\).
Second bracket: \(x^2 - y\).
\[ \text{Value} = (1 - y + x^2) - (x^2 - y) = 1 \]
The result is the constant \(1\), so it depends on neither \(x\) nor \(y\).

Final Answer:
The triple product is \(1\) and depends on neither \(x\) nor \(y\), option (B). \[ \boxed{\text{neither } x \text{ nor } y} \]