Question

Let $a=i-k,b=xi+j+(1-x)k$ and $c=yi+xj+(1+x-y)k$. Then, $\left[\begin{array}{ccc}a& b& c\end{array}\right]$ depends on

• A. Neither $x$ nor $y$
• B. Both $x$ and $y$
• C. Only $x$
• D. Only $y$

Answer 1

The correct option is A Neither $x$ nor $y$

Given, $a=i-k,b=xi+j+(1-x)k$ and $c=yi+xj+(1+x-y)k$
Therefore, $\left[\begin{array}{ccc}a& b& c\end{array}\right]=\left|\begin{array}{ccc}1& 0& -1\\ x& 1& 1-x\\ y& x& 1+x-y\end{array}\right|$
Applying $C_3\to C_3+C_1$
$=\left|\begin{array}{ccc}1& 0& 0\\ x& 1& 1\\ y& x& 1+x\end{array}\right|$
$=1[1+x-x]=1$
Thus, $\left[\begin{array}{ccc}a& b& c\end{array}\right]$ does not depend upon neither $x$ nor $y$.