A) \[[\mathbf{r}\,\ {{\mathbf{a}}_{1}}\ \,{{\mathbf{a}}_{2}}]=0\]
B) \[[\mathbf{r}\ \,{{\mathbf{a}}_{1}}\ \,{{\mathbf{a}}_{2}}]={{\mathbf{a}}_{1}}.\ {{\mathbf{a}}_{2}}\]
C) \[[\mathbf{r}\ \,{{\mathbf{a}}_{2}}\ \,{{\mathbf{a}}_{1}}]={{\mathbf{a}}_{1}}.\ {{\mathbf{a}}_{2}}\]
D) None of these
Correct Answer: A
Solution :
The required plane passes through a point having position vector \[{{\mathbf{a}}_{1}}\] and is parallel to the vectors \[{{\mathbf{a}}_{1}}\] and \[{{\mathbf{a}}_{2}}\]. If \[\mathbf{r}\] is the position vector of any point on the plane, then \[\mathbf{r}-{{\mathbf{a}}_{1}},{{\mathbf{a}}_{1}},{{\mathbf{a}}_{2}}\] are coplanar. Therefore, \[(\mathbf{r}-{{\mathbf{a}}_{1}}).({{\mathbf{a}}_{1}}\times {{\mathbf{a}}_{2}})=0\] Þ \[[\mathbf{r}\,\,{{\mathbf{a}}_{1}}\,{{\mathbf{a}}_{2}}]\]=\[[{{\mathbf{a}}_{1}}\,{{\mathbf{a}}_{1}}\,{{\mathbf{a}}_{2}}]\Rightarrow [\mathbf{r}\,{{\mathbf{a}}_{1}}\,{{\mathbf{a}}_{2}}]=0\] Hence, the required plane is \[[\mathbf{r}\,\,{{\mathbf{a}}_{1}}\,{{\mathbf{a}}_{2}}]=0\].
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