The following is a section from Educative Commentary on JEE 2008 - Mathematics - by K.D.Joshi, which is an optional internet resource* for students preparing for the JEE (India). The following image is a solution for a problem which needs collinearity of three points to be satisfied:
I do not have any trouble in understanding Statement-(2) which directly follows from the fact that "If three points are collinear then the area formed by the points will be zero".
It would be great if you could explain what is the meaning of Statement-(1) in the above image? How can the same rule be applied to $n$-dimensional space (I am interested only till $3$-dimensions)? How does varying the value of $\lambda$ determine the position of point $C$?
*Click here and see page 49, in case you want to access the original source of the image.
It comes from $$\overrightarrow{AC}=\lambda\overrightarrow{AB}$$