Warm-up

Find the distance between two points: A(6, -7) and B (4,8)

Find the distance between two points:C(3, 5, -6) and D(4, -6, 9)

€

229 or ≈15.1

€

347 or ≈18.6

Space Coordinates

In this lesson you will learn:

• 3 Space - The three-dimensional coordinate system • Points in space, ordered triples• The distance between two points in space• The midpoint between two points in space

• http://hotmath.com/learning_activities/interactivities/3dplotter.swf

Let’s look at 3-dimensional space.

To construct a 3-dimensional system, start with a yz plane flat on the paper(or screen).

y

zNext, the x-axis is perpendicularthrough the origin. (Think of the x-axis as coming out of the screen towards you.)

For each axis drawn the arrow represents the positive end.

x

Three-Dimensional Space

y

z

x

This is considered a right-handed system.

To recognize a right-handed system, imagine your right thumb pointing up the positive z-axis, your fingers curl from the positive x-axis to the positive y-axis.

In a left-handed system, if your left thumb is pointing up the positive z-axis, your fingers will still curl from the positive x-axis to the positive y-axis. Below is an example of a left-handed system.

z

x

y

Throughout this lesson, we will use right-handed systems.

x

y

z

The 3-dimensional coordinate system is divided into eight octants. Three planes shown below separate 3 space into the eight octants.

The three planes are the yz plane which is perpendicular to the x-axis, the xy plane which is perpendicular to the z-axis and the xz plane which is perpendicular to the y-axis.

Think about 4 octants sitting on top of the xy plane and the other 4 octants sitting below the xy plane.

yz plane

x

y

z

xy plane

x

y

z

xz plane

Every position or point in 3-dimensional space is identified by an ordered triple,(x, y, z).

Here is one example of plotting points in 3-dimensional space:

Plotting Points in Space

y

z

P (3, 4, 2)

The point is 3 units in front of the yz plane,4 points in front of the xz plane and 2 units up from the xy plane.

x

Here is another example of plotting points in space. In plotting the point Q (-3,4,-5) you will need to go back from the yz plane 3 units, out from the xz plane 4 units and down from the xy plane 5 units.

y

z

Q (-3, 4, -5)

As you can see it is more difficult to visualize points in 3 dimensions.

x

Distance Between Two Points in Space

The distance between two points

in space is given by the formula:

€

d = x2 − x1( )2

+ y2 − y1( )2

+ z2 − z1( )2

222111 ,,and,, zyxQzyxP

Find the distance between the points P(2, 3, 1) and Q(-3,4,2).

2.53327

1125

115

123423222

222

212

212

212

d

d

d

d

zzyyxxd

Example 1:

We will look at example problems related to the three-dimensional coordinate system as we look at the different topics.

Solution: Plugging into the distance formula:

Example 2:

Find the lengths of the sides of triangle with vertices (0, 0, 0), (5, 4, 1) and (4, -2, 3). Then determine if the triangle is a right triangle, an isosceles triangle or neither.

Solution: First find the length of each side of the triangle by finding thedistance between each pair of vertices.

(0, 0, 0) and (5, 4, 1)

42

11625

010405 222

d

d

d

(0, 0, 0) and (4, -2, 3)

29

9416

030204 222

d

d

d

(5, 4, 1) and (4, -2, 3)

41

4361

134254 222

d

d

d

These are the lengths of the sides of the triangle. Since none of them are equal we know that it is not an isosceles triangle and since we know it is not a right triangle. Thus it is neither.

€

42( )2

≠ 29( )2

+ 41( )2

The Midpoint Between Two Points in Space

The midpoint between two points, is given by:

222111 ,,and,, zyxQzyxP

2

,2

,2

Midpoint 212121 zzyyxx

Each coordinate in the midpoint is simply the average of the coordinatesin P and Q.

€

Solution :2 + −4

2,3+ 4

2,0 + 2

2

⎛

⎝ ⎜

⎞

⎠ ⎟=

−2

2,7

2,2

2

⎛

⎝ ⎜

⎞

⎠ ⎟= −1,

7

2,1

⎛

⎝ ⎜

⎞

⎠ ⎟

Example 3: Find the midpoint of the points P(2, 3, 0) and Q(-4,4,2).