Question

$\overline{a}=\overline{i}-\overline{k},\overline{b}=x\overline{i}+\overline{j}+(1-x)\overline{k}$ and $\overline{c}=y\overline{i}+\lambda \overline{j}+(1+x-y)\overline{k}$ then $\left[\overline{a}\overline{b}\overline{c}\right]$ depends on:-

• A. neither x nor y
• B. both x and y
• C. only x
• D. only y

Answer 1

Answer: (C)

only x

$\overline{a}.[\overline{b}\times \overline{c}]$

$\left|\begin{array}{ccc}1& 0& -1\\ x& 1& 1-x\\ y& \lambda & 1+x-y\end{array}\right|$

$⟹1(1+x-y-\lambda +\lambda x)-1(\lambda x-y)$

$⟹1+x-y-\lambda +\lambda x-\lambda x+y$

$\therefore \left[\begin{array}{ccc}a& b& c\end{array}\right]$ depends only on x,not y.