Identifying positions using three coordinates | Year 8

01 / LEARN

Understand the idea.

Three coordinates, one location

An ordered triple $(x, y, z)$ records one position in three dimensions. When the axes mean across, depth and height, read those values in that order. From the origin, move along $x$ , then parallel to $y$ , then parallel to $z$ to locate the point.

The coordinate order is part of the address. In the warehouse model, $(3, 2, 4)$ means across 3, depth 2 and height 4; $(3, 4, 2)$ describes a different location. Read the labelled axes before interpreting a slanted drawing.

Labelled x, y and z axes have numbered one-unit steps. A dashed gold route reaches P by following the x direction, then the y direction, then the z direction. The perspective projection does not preserve spatial lengths.

Zero and negative coordinates

A zero coordinate locates a point on a coordinate plane: $z = 0$ gives the $x y$ -plane, $x = 0$ gives the $y z$ -plane, and $y = 0$ gives the $x z$ -plane. Two zero entries place the point on an axis; $(0, 0, 0)$ is the origin.

A negative entry indicates the opposite of that axis's positive direction. For example, $(- 2, 3, 1)$ is 2 units in negative $x$ , 3 in positive $y$ and 1 in positive $z$ . Each entry controls only its own direction.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Read a shelf position

In a model warehouse, x is across, y is depth and z is shelf height. Describe P(3,2,4).

Worked solution

1
Identify what each coordinate measures.
$(x, y, z) = (across, depth, height)$
The question specifies the coordinate convention for this warehouse.
2
Read the entries without rearranging them.
$x = 3, y = 2, z = 4$
An ordered triple assigns each number to its fixed coordinate position.
3
Describe the shelf position.
P is across 3 units, at depth 2 units and at height 4 units; swapping depth and height would identify a different position.

Answer

Across 3, depth 2, height 4.

EXAMPLE 02

A point on the floor

Which of A(2,1,0), B(0,2,3) and C(1,0,2) lies on the xy-plane?

Worked solution

1
State the condition for the xy-plane.
$z = 0$
Every point in this plane has zero height.
2
Inspect the third entry of each point.
$z_{A} = 0, z_{B} = 3, z_{C} = 2$
The third entry is z, regardless of any zero in another position.
3
Select the point satisfying the condition.
$A = (2, 1, 0)$
Only A has z equal to zero, so only A lies on the xy-plane.

Answer

A only.

03 / YOUR TURN

Try it yourself.

Check your understanding

What is the height of (4,2,5) if z measures height?

Show answer & reasoning

5 units.

The third entry measures height. For $(4, 2, 5)$ , $z = 5$ , so the point's height is 5 units.

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

Read the $x$ -, $y$ - and $z$ -coordinates of $(2, 5, 1)$ .
2 marks
A drone is 4 units across, 2 units back and 3 units high. Write its position as $(x, y, z)$ .
2 marks
Which is on the floor $z = 0$ : $A (1, 0, 2)$ , $B (3, 4, 0)$ or $C (0, 2, 1)$ ?
2 marks
Describe $(- 2, 1, 3)$ using the positive axis directions as reference.
2 marks
Are $(2, 3, 4)$ and $(2, 4, 3)$ the same point? Explain.
2 marks
Which axis contains $(0, 0, 5)$ ?
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Read the $x$ -, $y$ - and $z$ -coordinates of $(2, 5, 1)$ .
2 marks
Read entries from first to third.
$x = 2$ , $y = 5$ , $z = 1$ .
A drone is 4 units across, 2 units back and 3 units high. Write its position as $(x, y, z)$ .
2 marks
Across is $x$ , depth is $y$ and height is $z$ .
$(4, 2, 3)$
Which is on the floor $z = 0$ : $A (1, 0, 2)$ , $B (3, 4, 0)$ or $C (0, 2, 1)$ ?
2 marks
$B$ has third coordinate zero.
$B$ .
Describe $(- 2, 1, 3)$ using the positive axis directions as reference.
2 marks
The minus sign belongs only to $x$ .
2 units in negative $x$ , 1 in positive $y$ and 3 in positive $z$ .
Are $(2, 3, 4)$ and $(2, 4, 3)$ the same point? Explain.
2 marks
The entries have a fixed order; swapping entries changes the location.
No. They have different $y$ - and $z$ -coordinates.
Which axis contains $(0, 0, 5)$ ?
2 marks
Both horizontal entries are zero; only $z$ varies.
The $z$ -axis.

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics

Coordinate transformations review; identifying positions using three coordinates; interpreting 3D axes; describing routes and relative positions; comparing diagrams, views and digital representations. This is spatial location, not senior vector algebra.

Official curriculum source ↗

In this lesson

Understand the idea.

Three coordinates, one location

Zero and negative coordinates

See the method in action.

Read a shelf position

Identify what each coordinate measures.

Read the entries without rearranging them.

Describe the shelf position.

A point on the floor

State the condition for the xy-plane.

Inspect the third entry of each point.

Select the point satisfying the condition.

Try it yourself.

Check your understanding

Keep practising