Question
If $$\vec a,\,\vec b,\,\vec c$$ are non-coplanar vectors and $$\lambda $$ is a real number, then the vectors $$\vec a + 2\vec b + 3\vec c,\,\lambda \vec b + 4\vec c$$ and $$\left( {2\lambda - 1} \right)\vec c$$ are non coplanar for :
A.
no value of $$\lambda $$
B.
all except one value of $$\lambda $$
C.
all except two values of $$\lambda $$
D.
all values of $$\lambda $$
Answer :
all except two values of $$\lambda $$
Solution :
Vectors $$\vec a + 2\vec b + 3\vec c,\,\lambda \vec b + 4\vec c$$ and $$\left( {2\lambda - 1} \right)\vec c$$ are coplanar if \[\left| \begin{array}{l}
1\,\,\,\,\,2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,3\\
0\,\,\,\,\lambda \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,4\\
0\,\,\,\,0\,\,\,\,\,\,2\lambda - 1
\end{array} \right| = 0\]
$$\eqalign{
& \Rightarrow \lambda \left( {2\lambda - 1} \right) = 0 \cr
& \Rightarrow \lambda = 0\,\,{\text{or}}\,\,\frac{1}{2} \cr} $$
$$\therefore $$ Forces are noncoplanar for all $$\lambda ,$$ except $$\lambda = 0,\,\frac{1}{2}$$