MATH1110 Lecture Notes - Lecture 10: Cartesian Coordinate System, Dot Product
18 Jun 2018
School
Department
Course
Professor
MATH1110: Mathematics 1
Vectors (IV): Lines and Planes
1 Introduction
•A line is determined once we know two points that lie on it.
•The (Cartesian) equation of a straight line in R2can be written as
ax +by +c= 0
•A plane is determined once we know three points that lie on it.
•The (Cartesian) equation of a plane in R3can be written as
ax +by +cz +d= 0
•The equations of a line in R3are
a1x+b1y+c1z+d1= 0
a2x+b2y+c2z+d2= 0
(i.e. the set of points that satisfy both of these equations is a line in
R3.)
•An alternative way of writing a line in Cartesian form is to use equations
that look like: x−x0
a=y−y0
b=z−z0
c
It is an annoying fact that there is no single Cartesian equation of a
line in 3 dimensions.
Page 1
Document Summary
R3. : an alternative way of writing a line in cartesian form is to use equations that look like: x x0 a y y0 b z z0 c. It is an annoying fact that there is no single cartesian equation of a line in 3 dimensions. Sketch the line in r3 given by the equations x = 1, y = 1. Sketch the line in r3 given by the equations x + y z = 2. Find equations for the line passing through the points (1, 1, 2) and (2, 1, 1). These can be written as a1x + b1y + c1z + d1 = 0 a2x + b2y + c2z + d2 = 0. Since (1, 1, 2) lies on both planes. And since (2, 1, 1) lies on both planes. From these two sets of equations we get. Choosing values for c1 and c2 will give us the required equations.
