If the points $\left(\lambda, 0, 3\right)$, $\left(1, 3 , -1\right)$ and $\left(-5, -3, 7\right)$ are collinear, then find $\lambda$.
Let $\overrightarrow{a}$, $\overrightarrow{b}$ and $\overrightarrow{c}$ be the position vectors of the points $\left(\lambda, 0, 3\right)$, $\left(1, 3 , -1\right)$ and $\left(-5, -3, 7\right)$ respectively.
Then,
$\overrightarrow{a} = \lambda \hat{i} + 3 \hat{k}$
$\overrightarrow{b} = \hat{i} + 3 \hat{j} - \hat{k}$
$\overrightarrow{c} = -5 \hat{i} - 3 \hat{j} + 7 \hat{k}$
$\because$ $\;$ The points are collinear, the position vectors of the points are coplanar.
The position vectors are coplanar $\implies$ $\left[\overrightarrow{a} \; \overrightarrow{b} \; \overrightarrow{c}\right] = 0$
Now,
$\left[\overrightarrow{a} \; \overrightarrow{b} \; \overrightarrow{c}\right] = 0$ $\implies$ $\begin{vmatrix}
\lambda & 0 & 3 \\
1 & 3 & -1 \\
-5 & -3 & 7
\end{vmatrix} = 0$
i.e. $\;$ $\lambda \left(21 - 3\right) + 3 \left(- 3 + 15\right) = 0$
i.e. $\;$ $18 \lambda = -36$ $\implies$ $\lambda = -2$