Mathematics - Vector Question with Solution | TestHub

$\vec{a}=\hat{i}-\hat{k},\vec{b}=x\hat{i}+\hat{j}+(1-x)\hat{k}$ and $\vec{c}=y\hat{i}+x\hat{j}+(1+x-y)\hat{k}$ $[\vec{a}\vec{b}\vec{c}]=\vec{a}\cdot (\vec{b}\times \vec{c})$ $$\begin{array}{rl}& \vec{b}\times \vec{c}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ x& 1& 1-x\\ y& x& 1+x-y\end{array}\right|=\hat{i}\left(1+x-x-x^2\right)-\hat{j}\left(x+x^2-xy-y+xy\right)+\hat{k}\left(x^2-y\right)\\ & \vec{a}.(\vec{b}\times \vec{c})=1\end{array}$$ which does not depend on $\mathrm{x}$ and $\mathrm{y}$