276 CHAPTER 13. ALGEBRA AND GEOMETRY OF Rp

Definition 13.4.1 A line in Rp containing the two different points x1 and x2 is thecollection of points of the form

x= x1 + t(x2 −x1)

where t ∈ R. This is known as a parametric equation and the variable t is called theparameter.

Often t denotes time in applications to Physics. Note this definition agrees with theusual notion of a line in two dimensions and so this is consistent with earlier concepts.

Lemma 13.4.2 Let a,b ∈ Rp 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(x2 −x1

)and so x= ta+b 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 forthe line.

Definition 13.4.4 Let p and q be two points in Rp, p ̸= q. The directed line seg-ment from p to q, denoted by −→pq, is defined to be the collection of points

x= p+ t (q−p) , t ∈ [0,1]

with the direction corresponding to increasing t. In the definition, when t = 0, the point p isobtained 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 increasingt.

Think of −→pq as an arrow whose point is on q and whose base is at p as shown in thefollowing picture.

q

p

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 (1,2,0) and(2,−4,6) .

Use the definition of a line given above to write

(x,y,z) = (1,2,0)+ t (1,−6,6) , t ∈ R.

The vector (1,−6,6) is obtained by (2,−4,6)− (1,2,0) 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 youobtain 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 isthis: The line is a set of points but the parametric description gives more information thanthat. It tells how the points are obtained. Obviously, there are many ways to trace out agiven set of points and each of these ways corresponds to a different parametric equationfor the line.