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Unit G1 Section 6
Three-Dimensional Coordinates

Coordinates can also be applied to three dimensions. Three numbers are required to identify the position of a point in space.

The coordinates are then written as ordered triples in the form of ( x, y, z ).

Worked Examples

1

The following diagram shows a cuboid.

Write down the coordinates of each vertex of the cuboid.

A (0, 0, 0)

B (8, 0, 0)

C (8, 4, 0)

D (0, 4, 0)

E (0, 0, 3)

F (8, 0, 3)

G (8, 4, 3)

H (0, 4, 3)
2

The diagram shows a triangular-based pyramid, ABCD, in which D is vertically above A. The base ABC is horizontal. Write down the coordinates of each of the vertices.

A (1, 1, 0)

B (6, 1, 0)

C (2, 6, 0)

D (1, 1, 4)
3

The diagram shows a cuboid and the x, y and z axes.

If AB = 6, AD = 5 and AE = 2 , write down the coordinates of each vertex of the cuboid.

A (0, 0, 0)

D (0, 5, 0)

G (6, 5, 2)

B (6, 0, 0)

E (0, 0, 2)

H (0, 5, 2)

C (6, 5, 0)

F (6, 0, 2)

Exercises

Write down the coordinates of each of the vertices of the following cuboids.

(a)

A (, , )

B (, , )

C (, , )

D (, , )

E (, , )

F (, , )

G (, , )

H (, , )

(b)

I (, , )

J (, , )

K (, , )

L (, , )

M (, , )

N (, , )

O (, , )

P (, , )

(c)

Q (, , )

R (, , )

S (, , )

T (, , )

U (, , )

V (, , )

W (, , )

X (, , )

(d)

A (, , )

B (, , )

C (, , )

D (, , )

E (, , )

F (, , )

G (, , )

H (, , )

Write down the coordinates of each of the vertices of the following solids.

(a)

A (, , )

B (, , )

C (, , )

D (, , )

(b)

D (, , )

E (, , )

F (, , )

G (, , )

(c)

H (, , )

I (, , )

J (, , )

K (, , )

L (, , )

(d)

M (, , )

N (, , )

O (, , )

P (, , )

Q (, , )

R (, , )

The diagram shows a tetrahedron.

If AB = 3, AC = 4 and AD = 5, write down the coordinates of each vertex of the tetrahedron.

A (, , )

B (, , )

C (, , )

D (, , )

In the diagram below, the x- and y-axes are horizontal and the z-axis is vertical. The diagram shows a triangular prism with a horizontal base, ABCD. The faces of the prism, AED and BFC, are vertical.

If AB = 7, AD = 4 and DE = 5, write down the coordinates of each vertex of the prism.

A (, , )

B (, , )

C (, , )

D (, , )

E (, , )

F (, , )

In the diagram below, the x- and y-axes are horizontal and the z-axis is vertical. The diagram shows a triangular prism with horizontal faces, ABC and DEF. The edges BE and CF are vertical.

If AB = 4, AC = 5 and AD = 8, write down the coordinates of each vertex of the prism.

A (, , )

B (, , )

C (, , )

D (, , )

E (, , )

F (, , )