Here are some questions we will address.
How do we describe a line in 2 dimensions or a line or plane in 3?
How can you find a point on a line or in a plane?
The distance between a point and a plane, or a point and a line?
The distance between two skew lines in 3D space?
The distance between parallel lines?
The projection of a point onto a line or plane?
All of these can be answered by appropriate definition and manipulation of vectors.