108. Coordinates in Three-Dimensional Space

Learning Intentions

• To know that the position of objects in three-dimensional (3D) space can be described using 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 $x$-coordinate and $y$-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 $(4,2,5)$:

a) the $x$-coordinate
b) the $y$-coordinate
c) the $z$-coordinate

Worked Example 2

Describe the position of each point in 3D space:

a) $A(3,1,2)$
b) $B(0,4,5)$
c) $C(2,0,1)$

Worked Example 3

List the coordinates of the point:

a) $P$ is $5$ units along $x$, $2$ units along $y$, and $3$ units up $z$
b) $Q$ is at the origin in $x$, $6$ units along $y$, and $1$ unit up $z$

Worked Example 4

Find which points lie on the same plane:

a) which point lies on the plane $z=0$: $A(2,3,0)$ or $B(2,3,4)$
b) which point lies on the plane $x=0$: $C(0,5,1)$ or $D(4,5,1)$

Worked Example 5

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

a) $A(2,1,4)$ and $B(2,1,9)$
b) $C(5,3,2)$ and $D(1,3,2)$

Worked Example 6

Find the midpoint of each line segment:

a) from $A(2,4,6)$ to $B(6,4,6)$
b) from $C(1,3,5)$ to $D(1,7,5)$
c) from $E(4,2,8)$ to $F(4,2,2)$

Problems

Problem 1

State what each coordinate represents in the point $(6,3,1)$:

a) the $x$-coordinate
b) the $y$-coordinate
c) the $z$-coordinate

Problem 2

Describe the position of each point in 3D space:

a) $A(2,5,1)$
b) $B(4,0,3)$
c) $C(1,2,0)$

Problem 3

List the coordinates of the point:

a) $P$ is $4$ units along $x$, $3$ units along $y$, and $2$ units up $z$
b) $Q$ is at the origin in $x$, $2$ units along $y$, and $7$ units up $z$

Problem 4

Find which points lie on the same plane:

a) which point lies on the plane $z=0$: $A(3,1,0)$ or $B(3,1,5)$
b) which point lies on the plane $y=0$: $C(4,0,2)$ or $D(4,6,2)$

Problem 5

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

a) $A(3,2,1)$ and $B(3,2,8)$
b) $C(7,4,5)$ and $D(2,4,5)$

Problem 6

Find the midpoint of each line segment:

a) from $A(1,5,3)$ to $B(7,5,3)$
b) from $C(4,2,6)$ to $D(4,8,6)$
c) from $E(6,1,9)$ to $F(6,1,3)$

Exercises

Understanding and Fluency

1. Write the coordinates of each point:
   a) $x=2,y=3,z=4$
   b) $x=5,y=0,z=1$
   c) $x=0,y=6,z=2$

2. State the value of each coordinate:
   a) for $A(4,1,7)$, state $x,y,z$
   b) for $B(0,3,5)$, state $x,y,z$
   c) for $C(6,2,0)$, state $x,y,z$

3. Describe the position of each point in words:
   a) $P(3,4,2)$
   b) $Q(1,0,5)$
   c) $R(0,2,1)$

4. Decide which point lies on each plane:
   a) plane $x=0$: $A(0,3,4)$ or $B(2,3,4)$
   b) plane $y=0$: $C(5,0,1)$ or $D(5,4,1)$
   c) plane $z=0$: $E(6,2,0)$ or $F(6,2,3)$

5. Find the distance between each pair of points:
   a) $(2,3,4)$ and $(2,3,9)$
   b) $(7,1,5)$ and $(3,1,5)$
   c) $(4,6,2)$ and $(4,2,2)$

6. Find the midpoint of each line segment:
   a) $(2,1,3)$ and $(6,1,3)$
   b) $(5,4,7)$ and $(5,8,7)$
   c) $(3,2,9)$ and $(3,2,1)$

7. Which coordinate changes?
   a) from $(1,2,3)$ to $(1,2,8)$
   b) from $(4,5,6)$ to $(4,1,6)$
   c) from $(7,3,2)$ to $(2,3,2)$

8. Determine whether each statement is true or false:
   a) A 3D coordinate has three numbers
   b) The point $(0,0,0)$ is the origin
   c) Points with the same $z$-coordinate are always the same point
   d) A point on the plane $z=0$ has no height above the base plane

Reasoning

9. Explain why $(3,2,4)$ and $(2,3,4)$ are different points.

10. A student says that the point $(5,0,2)$ cannot exist because one coordinate is zero. Explain the mistake.

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

12. Explain why the midpoint of the segment from $(2,4,6)$ to $(8,4,6)$ is halfway only in the $x$-direction.

13. A student says that any point with $z=0$ is not in 3D space. Describe the error.

Problem-solving

14. A drone is at point $(4,3,7)$. Describe its position in 3D space.

15. Two storage boxes are at $(2,5,1)$ and $(2,5,6)$. How far apart are they?

16. A point moves from $(1,4,2)$ to $(1,9,2)$. Which direction has changed, and by how many units?

17. A line segment joins $(3,6,4)$ and $(9,6,4)$. Find its midpoint.

18. A point lies on the plane $x=0$ and has coordinates $(0,7,3)$. Explain what this tells you about its position.

19. Three points are $A(2,1,5)$, $B(2,6,5)$ and $C(2,6,1)$. Find the distances $AB$ and $BC$.

Potential Misunderstandings

• Students may think a 3D coordinate can be written in any order
• Students may confuse the roles of the $x$-, $y$- and $z$-coordinates
• Students may think a coordinate containing $0$ 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 $(0,0,0)$
• Students may think the plane $z=0$ 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 $x=0$ with a line