Question

Let $\vec{a} = \hat{i} - \hat{k} , \vec{b} = x \hat{i} + \hat{j} + (1 - x) \hat{k}$ and $\vec{c} = y \hat{i} + x \hat{j} + ( 1 +x -y ) \hat{k} . $ Then $[\vec{a} , \vec{b} , \vec{c} ] $ depends on

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

Answer 1

Answer: (D)

$\vec{a} = \hat{i} - \hat{k} , \vec{b} = x \hat{i} + \hat{j} + \left(1-x\right)\hat{k} $
and $ \vec{c} = y \hat{i} + x \hat{j} + \left(1+ x -y\right) \hat{k} $
$ \left[\vec{a} \vec{b} \vec{c}\right] = \vec{a} \vec{b} \times\vec{c} = \begin{vmatrix}1&0&-1\\ x&1&1-x\\ y&x&1+x-y\end{vmatrix} $
$ = 1\left[1+x-y-x+x^{2}\right] - \left[x^{2} -y\right]$
$ = 1 - y + x^{2} -x^{2} + y $
$ = 1 $
Hence $[\vec{a} ,\vec{b} ,\vec{c} ] $ is independent of $x$ and $y$ both.