To begin with consider the case n = 1,2. In the case where n = 1, the only line is just
ℝ1 = ℝ. Therefore, if x1 and x2 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
be two different
which are contained in a line l.
Suppose that x1≠x2.
arbitrary point on
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
If x1 = x2, then in place of the point slope form above, x = x1. Since the two given points
are different, y1≠y2 and so you still obtain the above formula for the line. Because of this,
the following is the definition of a line in ℝn.
Definition 13.4.1 A line in ℝn containing the two different points x1 and x2 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
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
Lemma 13.4.2 Let a,b ∈ ℝn with a≠0. Then x = ta + b,t ∈ ℝ, is a line.
Proof: Let x1 = b and let x2 −x1 = a so that x2≠x1. Then ta + b = x1 + t
is a line containing the two different points x1
and x2. ■
Definition 13.4.3 The vector a in the above lemma is called a direction
vector for the line.
Definition 13.4.4 Let p and q be two points in ℝn, p≠q. The directed
line segment from p to q, denoted by
, is defined to be the collection of
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
as an arrow whose point is on
and whose base is at p
as shown in the
This line segment is a part of a line from the above Definition.
Example 13.4.5 Find a parametric equation for the line through the points
Use the definition of a line given above to write
is obtained by
as indicated above.
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
Example 13.4.6 Find a parametric equation for the line which contains the point
and has direction vector
From the above this is just
Sometimes people elect to write a line like the above in the form
This is a set of scalar parametric equations which amounts to the same thing as
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
This is the symmetric form of the line.
Example 13.4.7 Suppose the symmetric form of a line is
Find the line in parametric form.
Let t =
Then solving for x,y,z,
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.