question_answer

  Let the volume of a parallelepiped whose coterminous edges are given by \[\vec{u}=\vec{i}+\vec{j}+\lambda \hat{k},\]\[\vec{v}=\hat{i}+\hat{j}+3\hat{k}\] and \[\vec{w}=2\hat{i}+\hat{j}+\hat{k}\] be 1 cu. Unit if \[\theta \] be the angle between the edges \[\vec{u}\] and \[\vec{w}\], then \[\cos \theta \]can be [JEE MAIN Held On 08-01-2020 Morning]

A) \[\frac{5}{7}\]              

B) \[\frac{5}{3\sqrt{3}}\]

C) \[\frac{7}{6\sqrt{6}}\]   

D) \[\frac{7}{6\sqrt{3}}\]

Correct Answer: D

Solution :

[d] \[v=\left[ \vec{a}\vec{b}\vec{c} \right]\] \[\Rightarrow 2-1(5)+\lambda (1)=\pm 1\] \[\Rightarrow \lambda -3=1\] or \[\lambda -3=-1\] \[\Rightarrow \lambda =4\,\,\text{or}\,\,2\] \[\vec{u}=\hat{i}+\hat{j}+4\hat{k}\] or \[\hat{i}+\hat{j}+2\hat{k}\] \[\therefore \cos \theta =\frac{2+1+4}{\sqrt{18}\sqrt{6}}or\frac{2+1+2}{\sqrt{6}\sqrt{6}}\] \[=\frac{7}{6\sqrt{3}}\,\,\text{or}\,\,\frac{5}{6}\]