question_answer

If the vectors, \[\mathbf{\vec{a}}=\mathbf{2\hat{i}}-\mathbf{3\hat{J}}+\mathbf{4\hat{k}},\mathbf{\hat{b}}=\mathbf{\hat{i}}+\mathbf{2\hat{J}}-\mathbf{\hat{k}}\]and \[\mathbf{\vec{c}}=\mathbf{x\hat{i}}-\mathbf{\hat{j}}+\mathbf{2\hat{k}}\] are coplanar, then x=

A)   0                    

B)  1              

C)  \[\frac{5}{8}\]                         

D)  \[\frac{8}{5}\]

Correct Answer: D

Solution :

[d] \[\because \vec{a},\vec{b}\,\,and\,\,\vec{c}\]are coplanar. \[\Rightarrow 2\left( 4-1 \right)-1\left( -6+4 \right)+x\left( 3-8 \right)=0\] \[2\times 3-1\times \left( -2 \right)+\times \left( -5 \right)=0\] \[6+2-5x=0\] \[5x=8\] \[\therefore x=\frac{8}{5}\] Hence, option [d] is correct.