Question
Let $$\vec a = \vec i - \vec k,\,\vec b = x\vec i + \vec j + \left( {1 - x} \right)\vec k$$ and $$\vec c = y\vec i + x\vec j + \left( {1 + x - y} \right)\vec k.$$ Then $$\left[ {\vec a\,\vec b\,\vec c} \right]$$ depends on :
A.
only $$x$$
B.
only $$y$$
C.
neither $$x$$ nor $$y$$
D.
both $$x$$ and $$y$$
Answer :
neither $$x$$ nor $$y$$
Solution :
\[\begin{array}{l}
\vec a = \hat i - \hat k,\\
\vec b = x\hat i + \hat j + \left( {1 - x} \right)\hat k,\\
\vec c = y\hat i + x\hat j + \left( {1 + x - y} \right)\hat k\\
\left[ {\vec a\,\vec b\,\vec c} \right] = \left| \begin{array}{l}
1\,\,\,\,\,\,0\,\,\,\,\,\,\,\, - 1\\
x\,\,\,\,\,\,1\,\,\,\,\,\,\,1 - x\\
y\,\,\,\,\,\,x\,\,\,\,\,1 + x - y
\end{array} \right|\\
= 1\left( {1 + x - y - x + {x^2}} \right) - 1\left( {{x^2} - y} \right)\\
= 1
\end{array}\]
\[\therefore \] Depends neither on $$x$$ nor on $$y.$$