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

Since, $\bar{a}, \bar{b}$ and $\bar{c}$ are coplanar vectors.
$\Rightarrow\begin{bmatrix}\bar{a} & \bar{b} & \bar{c}\end{bmatrix}=0$
$\Rightarrow\begin{vmatrix}\lambda^{3} & 0 & 1 \\ 1 & -\lambda^{3} & 0 \\ 1 & 2 \lambda-\sin \lambda & -\lambda\end{vmatrix}=0$
$\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. $\geq 2$
$\Rightarrow$ No value of $\lambda$ satisfies (i)
$\Rightarrow$ Number of real value of $\lambda=1$