If the points $$A\left( {1,\,2,\, - 1} \right),\,B\left( {2,\,6,\,2} \right)$$     and $$C\left( {\lambda ,\, - 2,\, - 4} \right)$$   are collinear then $$\lambda $$ is :

Solution :
Direction ratios of the line $$AB$$  are $$2 - 1,\,6 - 2,\,2 - \left( { - 1} \right),$$     i.e., $$1,\,4,\,3.$$   Direction ratios of the line $$AC$$ are $$\lambda - 1,\, - 2 - 2,\, - 4 - \left( { - 1} \right),$$      i.e., $$\lambda - 1,\, - 4,\, - 3.$$    For collinearity, $$AB||AC\,\,\,\,\, \Rightarrow \frac{{\lambda - 1}}{1} = \frac{{ - 4}}{4} = \frac{{ - 3}}{3}\,\,\,\,\, \Rightarrow \lambda = 0.$$