LESSON 108Introduction to Coordinate Space

Three-dimensional coordinate system

The space is divided into 8 regions by the:• x-axis• y-axis• z-axis

Coordinate points are

ordered triples

(x, y, z)

We draw them as an

isometric perspective

Right-Hand Rule (“Thumb’s Up”)

Your index finger is the positive x-axis

Your arm is the positive y-axis

Your thumb is the positive z-axis

Coordinate Planes

The three coordinate axes determine the three coordinate planes• The xy-plane contains

the x- and y-axes.• The yz-plane contains

the y- and z-axes.• The xz-plane contains

the x- and z-axes.

Octants

These three coordinate planes divide space into 8 parts, called octants• The first octant, in the foreground, is determined by the positive axes

3-D COORDINATE SYSTEMS

Having some difficulty visualizing diagrams of 3-D figures?Then try standing in the corner.• The bottom corner of a

room is called the origin• The wall on your left is in

the xz-plane.• The wall on your right is in

the yz-plane.• The floor is in the xy-plane

3-D COORDINATE SYSTEMS

• The x-axis runs along the intersection of the floor and the left wall.

• The y-axis runs along that of the floor and the right wall.

• The z-axis runs up from the floor toward the ceiling along the intersection of the two walls.

3-D COORDINATE SYSTEMS

• You are situated in the first octant

• Can you now imagine seven other rooms situated in the other seven octants?• There are three on the

same floor and four on the floor below

• They are all connected by the common corner point O

Graph the triples (-4, 3, -5) & (3, -2, -6)

Finding collinear points

Determine the coordinates of two points that are collinear to the points R and S

We are going to use the idea of vectors to help with this problem

Subtract the coordinates of R and S to determine their direction of the vector

Replace t with a number other than 1 or 0: then add the coordinates

Determine if 3 points are collinear

Two birds fly from the point . One goes to point and the second goes to point . Are the birds flying in a collinear path?

No.

Since D is the start of both birds, do E minus D

Now see if a multiple of that vector takes you to point F

Questions?

When identifying coordinates, pay attention to other points in the same plane