Question

If the vectors $\bar{a}={\lambda }^3\hat{i}+\hat{k},\bar{b}=\hat{i}-{\lambda }^3\hat{j}$ and $\bar{c}=\hat{i}+(2\lambda -\sin \lambda )\hat{j}-\lambda \hat{k}$ are coplanar, then find the number of distinct real values of $\lambda$.

Answer 1

Answer: 1

Since, $\bar{a},\bar{b}$ and $\bar{c}$ are coplanar vectors.
$\Rightarrow \left[\begin{array}{ccc}\bar{a}& \bar{b}& \bar{c}\end{array}\right]=0$
$\Rightarrow \left|\begin{array}{ccc}{\lambda }^3& 0& 1\\ 1& -{\lambda }^3& 0\\ 1& 2\lambda -\sin \lambda & -\lambda \end{array}\right|=0$
$\Rightarrow {\lambda }^7+2\lambda -\sin \lambda +{\lambda }^3=0$
$\Rightarrow \lambda \left({\lambda }^6+{\lambda }^2+2\right)=\sin \lambda$
This is true for $\lambda =0$.
For non-zero values of $\lambda$,
${\lambda }^6+{\lambda }^2+2=\frac{\sin \lambda }{\lambda }...(i)$
R.H.S $<1$ and L.H.S. $\ge 2$
$\Rightarrow$ No value of $\lambda$ satisfies (i)
$\Rightarrow$ Number of real value of $\lambda =1$