If <span class="math-tex">\(\vec a = \hat i - \hat k,\; \vec b = x\hat i + \hat j + (1 - x)\hat k\)</span> and <span class="math-tex">\(c = y\hat i + x\hat j + (1 + x - y)\hat k,\)</span> then <span class="math-tex">\(\left[\vec a \vec b \vec c\right]\)</span> depends on
\([\vec{a}\,\vec{b}\,\vec{c}] = \begin{vmatrix} 1 & 0 & -1 \\ x & 1 & 1-x \\ y & x & 1+x-y \end{vmatrix}\)
\(= 1\left[(1+x-y) - x(1-x)\right] - 0 + (-1)\left[x^2 - y\right]\)
\(= 1 + x - y - x + x^2 - x^2 + y = 1\)
The value is always 1, so it depends on neither x nor y.
Answer: (A)
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