If the vectors a, b, c are co-planar, prove that the following determinant is zero.
Determinant first row: a, b, c
Determinant second row: a.a, a.b, a.c
Determinant third row: b.a, b.b, b.c
Note: Due to typing difficulty, the determinant is shown as above. Further, the vecor notation above a, b, c are not shown due to typing difficulty.
Since \(\vec a,\vec b,\vec c\) are coplanar, \(\vec c = \lambda \vec a + \mu \vec b\)
\(\left| {\begin{array}{*{20}{c}}
{\vec a}&{\vec b}&{\lambda \vec a + \mu \vec b}\\
{{a^2}}&{\vec a.\vec b}&{\lambda {a^2} + \mu \vec a.\vec b}\\
{\vec a.\vec b}&{{b^2}}&{\lambda \vec a.\vec b + \mu {b^2}}
\end{array}} \right|\)
Now, \({C_3} \to {C_3} - \lambda {C_1} - \mu {C_2}\)
PS: If you are not familiar with LaTeX, then there are other ways to present, like attaching scanned image, .doc file, open office document file etc. Not very refined but a workable way is to take picture using mobile phone/tablet and then attach the picture.