Two lines r = a _ 1 + b _ 1 and r = a _ 2 + b _ 2 would be coplanar if:
Mathematics · JEE Main · NTA Exams — Three Dimensional Geometry
Two lines \(\overrightarrow{\mathrm{r}}=\overrightarrow{\mathrm{a}}_{1}+\lambda \overrightarrow{\mathrm{b}}_{1} \text { and } \overrightarrow{\mathrm{r}}=\overrightarrow{\mathrm{a}}_{2}+\mu \overrightarrow{\mathrm{b}}_{2}\) would be coplanar if:
- \(\left[\vec{a}_{1} \vec{b}_{1} \vec{b}_{2}\right]=\left[\vec{a}_{2} \vec{b}_{1} \vec{b}_{2}\right]\)
- \(\left(\vec{a}_{1} \cdot \vec{b}_{1}\right) \vec{b}_{2}=\left(\vec{a}_{2} \cdot \vec{b}_{1}\right) \vec{b}_{2}\)
- \(\vec{a}_{1}\left(\vec{b}_{1} \cdot \vec{b}_{2}\right)=\vec{a}_{2}\left(\vec{b}_{1} \cdot \vec{b}_{2}\right)\)
- \(\vec{a}_{1} \cdot \vec{b}_{1}-\vec{a}_{1} \cdot \vec{b}_{2}=\vec{a}_{2} \cdot \vec{b}_{1}-\vec{a}_{2} \cdot \vec{b}_{2}\)
Answer
(A) [ a _ 1 b _ 1 b _ 2 ]= [ a _ 2 b _ 1 b _ 2 ]
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