Elementary Calculus: Theorem 1: The Line

The vector equation means that L is the set of all points X such that X = P + tD for some t.

PROOF

Let L be any line with position vector P and direction vector D. We must show that:

(i) L has the scalar equation given in Theorem 1.

(ii) If X = P + tD for some f then X is a point of L.

(iii) If X is a point of L then X = P + fD for some t.

(i) D is the vector from A to B where A and B are points on L. Since L is the line through A and B, it has the scalar equation

(x - a₁)(b₂ - a₂) = (y - a₂)(b₁ - a₁),

(x - a₁)d₂ = (y - a₂)d₁,

xd₂ - yd₁= a₁d₂ - a₂d₁.

This equation holds for the point P of L,

p₁d₂ - p₂d₁ = a₁d₂ - a₂d₁.

Combining the last two equations we get the required equation:

(1)

xd₂ - yd₁ = p₁d₂ - p₂d₁.

(ii) Let X be a point such that X = P + tD for some t. Then

x = p₁ + td₁, y = p₂ + td₂,

d₂x - d_xy = d₂p₁ + d₂td₁ - d₁p₂ - d₁td₂ = d₂p_x - d₁p₂,

so X is a point of L.

(iii) Let X be a point of L. If d₁ ≠ 0 we set t = (x - p₁/d₁ and using Equation 1 we get X = P + tD. The case d₂ = 0 is similar. Therefore X = P + tD is a vector equation for L (Figure 10.2.6).

Figure 10.2.6: The line with position vector P and direction vector D

In general, the line L through points A and B has the vector equation X = A + t(B - A) because A is a position vector and B - A is a direction vector of L.

A four-sided figure whose opposite sides represent equal vectors is called a parallelogram.