Question

If $a=i-k,b=xi+j+(1-x)k$ and $c=yi+xj+(1+x-y)k$. Then, $[abc]$ 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 $y$

Explanation for the correct option:

Step 1. Given that $a=i-k,b=xi+j+(1-x)k$ and $c=yi+xj+(1+x-y)k$

$\therefore$$\left[abc\right]=\left|\begin{array}{ccc}1& 0& -1\\ x& 1& 1-x\\ y& x& 1+x-y\end{array}\right|$

Step 2. Applying $C_3\to C_3+C_1$, we get

$\begin{array}{rcl}\left[abc\right]& =& \left|\begin{array}{ccc}1& 0& 0\\ x& 1& 1\\ y& x& 1+x\end{array}\right|\\ & =& 1\left[1+x-x\right]\\ & =& 1\end{array}$

$\therefore \left[abc\right]$ neither depend on $\mathit{x}$ nor $\mathit{y}$

Hence, Option ‘A’ is Correct.