A) only \[\lambda \]
B) only \[\mu \]
C) both \[\lambda \] and \[\mu \]
D) neither \[\lambda \] nor\[\mu \]
Correct Answer: D
Solution :
Given edges are \[\vec{a}=\hat{i}-\hat{k},\,\,\vec{b}=\lambda \hat{i}+\hat{j}+(1-\lambda )\hat{k}\] and \[\vec{c}=\mu \hat{i}+\lambda \hat{j}+(1+\lambda -\mu )\hat{k}\] \[\therefore \] Volume of parallelepiped \[=[\vec{a}\,\vec{b}\,\vec{c}]\] \[=\left| \begin{matrix} 1 & 0 & -1 \\ \lambda & 1 & 1-\lambda \\ \mu & \lambda & 1+\lambda -\mu \\ \end{matrix} \right|\] \[=1(1+\lambda -\mu -\lambda +{{\lambda }^{2}})-0-1({{\lambda }^{2}}-\mu )\] \[=1+{{\lambda }^{2}}-\mu -{{\lambda }^{2}}+\mu \] \[=1\] Hence, volume depends on neither \[\lambda \] nor \[\mu \].
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