58 CHAPTER 5. FUNDAMENTALS

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

y− y1 = m(x− x1) .

If t is defined byx = x1 + t (x2− x1) ,

you obtain this equation along with

y = y1 +mt (x2− x1)

= y1 + t (y2− y1) .

Therefore,(x,y) = (x1,y1)+ t (x2− x1,y2− y1) .

If x1 = x2, then in place of the point slope form above, x = x1. Since the two given pointsare 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 5.4.1 A line in Rn containing the two different points x1 and x2 is the collectionof 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 5.4.2 Let a,b ∈ Rn with a ̸= 0. Then x= ta+b ∈ 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 5.4.3 The vector a in the above lemma is called a direction vector for the line.

Definition 5.4.4 Let p and q be two points in Rn, p ̸= q. The directed line segment fromp 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.