108. Coordinates in Three-Dimensional Space

Learning Intentions

• To know that the position of objects in three-dimensional (3D) space can be described Use a 3D coordinate system
• Describe the position of an object or a point using a 3D coordinate system
• Solve simple geometry problems in 3D space using coordinates

Pre-requisite Summary

• Know that a coordinate gives the position of a point
• Understand how ordered pairs are used in a 2D coordinate system
• Know that the -coordinate and -coordinate describe horizontal and vertical position in 2D
• Understand that 3D space needs three coordinates instead of two
• Know that coordinates are written in order
• Be able to read points from labelled axes
• Understand that points can be compared by looking at each coordinate
• Be familiar with basic geometry language such as point, axis, distance and midpoint

Worked Examples

Worked Example 1

State what each coordinate represents in the point :

a) the -coordinate

b) the -coordinate

c) the -coordinate

Worked Example 2

Describe the position of each point in 3D space:

a)

b)

c)

Worked Example 3

List the coordinates of the point:

a) is units along , units along , and units up

b) is at the origin in , units along , and unit up

Worked Example 4

Solve which points lie on the same plane:

a) which point lies on the plane : or

b) which point lies on the plane : or

Worked Example 5

Find the distance between points that differ in only one coordinate:

a) and

b) and

Worked Example 6

Find the midpoint of each line segment:

a) from to

b) from to

c) from to

Problems

Problem 1

State what each coordinate represents in the point :

a) the -coordinate

b) the -coordinate

c) the -coordinate

Problem 2

Describe the position of each point in 3D space:

a)

b)

c)

Problem 3

List the coordinates of the point:

a) is units along , units along , and units up

b) is at the origin in , units along , and units up

Problem 4

Find which points lie on the same plane:

a) which point lies on the plane : or

b) which point lies on the plane : or

Problem 5

Find the distance between points that differ in only one coordinate:

a) and

b) and

Problem 6

Find the midpoint of each line segment:

a) from to

b) from to

c) from to

Exercises

Understanding and Fluency

Exercise 1.

Write the coordinates of each point:

a)

b)

c)

Exercise 2.

State the value of each coordinate:

a) for , state

b) for , state

c) for , state

Exercise 3.

Describe the position of each point in words:

a)

b)

c)

Exercise 4.

Decide which point lies on each plane:

a) plane : or

b) plane : or

c) plane : or

Exercise 5.

Find the distance between each pair of points:

a) and

b) and

c) and

Exercise 6.

Find the midpoint of each line segment:

a) and

b) and

c) and

Exercise 7.

Which coordinate changes?

a) from to

b) from to

c) from to

Exercise 8.

Determine whether each statement is true or false:

a) A 3D coordinate has three numbers

b) The point is the origin

c) Points with the same -coordinate are always the same point

d) A point on the plane has no height above the base plane

Reasoning

Exercise 9.

Explain why and are different points.

Exercise 10.

A student says that the point cannot exist because one coordinate is zero. Explain the mistake.

Exercise 11.

Noah says that if two points have the same - and -coordinates, then they must be the same point. Is he correct? Explain.

Exercise 12.

Explain why the midpoint of the segment from to is halfway only in the -direction.

Exercise 13.

A student says that any point with is not in 3D space. Describe the error.

Problem-solving

Exercise 14.

A drone is at point . Describe its position in 3D space.

Exercise 15.

Two storage boxes are at and . How far apart are they?

Exercise 16.

A point moves from to . Which direction has changed, and by how many units?

Exercise 17.

A line segment joins and . Find its midpoint.

Exercise 18.

A point lies on the plane and has coordinates . Explain what this tells you about its position.

Exercise 19.

Three points are , and . Find the distances and .

Potential Misunderstandings

• Students may think a 3D coordinate can be written in any order
• Students may confuse the roles of the -, - and -coordinates
• Students may think a coordinate containing is invalid
• Students may confuse a 2D coordinate pair with a 3D coordinate triple
• Students may think points with two matching coordinates must be the same point
• Students may read or write coordinates in the wrong order
• Students may forget that the origin in 3D is
• Students may think the plane means the point does not exist in 3D
• Students may try to Use full 3D distance methods when only one coordinate changes
• Students may find a midpoint by changing only one coordinate when more than one coordinate should be considered
• Students may confuse a plane such as with a line

Next: 109r. Equivalent Fractions and Simplifying Fractions