To begin with consider the case n = 1,2. In the case where n = 1, the only line is just ℝ^{1} = ℝ. Therefore, if x_{1} and x_{2} are two different points in ℝ, consider

where t ∈ ℝ and the totality of all such points will give ℝ. You see that you can always solve the above equation for t, showing that every point on ℝ is of this form. Now consider the plane. Does a similar formula hold? Let
Now by similar triangles,

and so the point slope form of the line, l, is given as

If t is defined by

you obtain this equation along with

Therefore,

If x_{1} = x_{2}, then in place of the point slope form above, x = x_{1}. Since the two given points are different, y_{1}≠y_{2} and so you still obtain the above formula for the line. Because of this, the following is the definition of a line in ℝ^{n}.
Definition 13.4.1 A line in ℝ^{n} containing the two different points x^{1} and x^{2} is the collection of points of the form

where t ∈ ℝ. This is known as a parametric equation and the variable t is called the parameter.
Often t denotes time in applications to Physics. Note this definition agrees with the usual notion of a line in two dimensions and so this is consistent with earlier concepts.
Proof: Let x^{1} = b and let x^{2} −x^{1} = a so that x^{2}≠x^{1}. Then ta + b = x^{1} + t
Definition 13.4.4 Let p and q be two points in ℝ^{n}, p≠q. The directed line segment from p to q, denoted by

with the direction corresponding to increasing t. In the definition, when t = 0, the point p is obtained and as t increases other points on this line segment are obtained until when t = 1, you get the point q. This is what is meant by saying the direction corresponds to increasing t.
Think of
Example 13.4.5 Find a parametric equation for the line through the points
Use the definition of a line given above to write

The vector
The reason for the word, “a”, rather than the word, “the” is there are infinitely many different parametric equations for the same line. To see this replace t with 3s. Then you obtain a parametric equation for the same line because the same set of points is obtained. The difference is they are obtained from different values of the parameter. What happens is this: The line is a set of points but the parametric description gives more information than that. It tells how the points are obtained. Obviously, there are many ways to trace out a given set of points and each of these ways corresponds to a different parametric equation for the line.
Example 13.4.6 Find a parametric equation for the line which contains the point
From the above this is just
 (13.11) 
Sometimes people elect to write a line like the above in the form
 (13.12) 
This is a set of scalar parametric equations which amounts to the same thing as 13.11.
There is one other form for a line which is sometimes considered useful. It is the so called symmetric form. Consider the line of 13.12. You can solve for the parameter t to write

Therefore,

This is the symmetric form of the line.
Let t =

Written in terms of vectors this is

I don’t understand why anyone would care about the symmetric form of a line if a parametric description is available. Indeed, in linear algebra, you do row operations to express the solution not as a symmetric equation but parametrically.