The angle theta between two vectors u and v is related to the inner product of u and v by the formula
Given a line x = a + t v in parametric form, where a is a point on the line and v a parallel vector, one may regard the parameter t as the coordinate of the point x in a (one-dimensional) coordinate system on the line.
Given a plane and the data consisting of a point a in the plane and two non-parallel vectors u, v in the plane, we may regard the parametric representation of the plane for the given data, namely,
as imposing a coordinate system on the plane in which the ordered pair (s, t) is considered the vector of coordinates of the point x.
Let L be the line in space passing through the point (1, -1, 2) and the point (2, 1, -3). Let t be the coordinate on this line assigning 0 to the first of these points and 1 to the second.
What point p on L has coordinate t = -2 ?
What point q on L has coordinate t = 3 ?
What is the distance in space from p to q ?
Let E be the plane in space with the parametric representation
Regard the parameter pair (s, t) as defining coordinates in this plane.
What are the vertices of the triangle in space that corresponds to the triangle with vertices (0, 1), (0, 0), and (1, 0) in the (s, t) plane?
Compare the angles and the lengths of the sides of the triangle in the (s, t) plane with those of the corresponding triangle in space.
Repeat the previous exercise when E is the plane