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12.1PureCore

3D Coordinates

Year 1 vectors were two-dimensional. In Year 2 we add a third axis, z, perpendicular to both x and y. Points and distances extend naturally to three dimensions.

22 min Vectors
Edexcel A-Level Maths: Vectors (Part 1)Curated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Use 3D coordinates (x, y, z)
  • Find the distance from the origin
  • Find the distance between two points
  • Visualise points in 3D space
1

The third axis

A point in space is written (x,y,z)(x, y, z), with the zz-axis perpendicular to the xyxy-plane. Distances use a 3D version of Pythagoras.

∣OP∣=x2+y2+z2|OP| = \sqrt{x^2 + y^2 + z^2}
Distance from the origin.
Example 1Find the distance from the origin to (2,3,6)(2, 3, 6).
122+32+62=4+9+36\sqrt{2^2 + 3^2 + 6^2} = \sqrt{4 + 9 + 36}.
2=49=7= \sqrt{49} = 7.
Answer
77
2

Distance between two points

The distance between (x1,y1,z1)(x_1, y_1, z_1) and (x2,y2,z2)(x_2, y_2, z_2) is (x2−x1)2+(y2−y1)2+(z2−z1)2\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}.

Tip — 3D distance is just Pythagoras with a third squared term added.

Formula recap

∣OP∣=x2+y2+z2|OP| = \sqrt{x^2 + y^2 + z^2}
From the origin.
(Δx)2+(Δy)2+(Δz)2\sqrt{(\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2}
Between two points.

Common mistakes to avoid

Forgetting the z-term in the distance formula.
Include all three squared differences.
Mixing up the order of coordinates.
Coordinates are always (x, y, z).

Key takeaways

  • Points in space are (x, y, z).
  • |OP| = √(x² + y² + z²).
  • Distance between points adds a third squared term.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. The distance from the origin to (x, y, z) is:

    • x2+y2+z2\sqrt{x^2+y^2+z^2}
    • x+y+zx+y+z
    • x2+y2\sqrt{x^2+y^2}
    • xyzxyz
    Show answer and explanation

    Answer: x2+y2+z2\sqrt{x^2+y^2+z^2}

    3D Pythagoras.

  2. The distance from the origin to (2, 3, 6) is:

    • 77
    • 1111
    • 41\sqrt{41}
    • 66
    Show answer and explanation

    Answer: 77

    √49.

  3. A point in 3D space is written:

    • (x, y, z)
    • (x, y)
    • (r, θ)
    • (x, z)
    Show answer and explanation

    Answer: (x, y, z)

    Three coordinates.

  4. The distance between (1,0,0) and (1,0,3) is:

    • 33
    • 11
    • 10\sqrt{10}
    • 00
    Show answer and explanation

    Answer: 33

    Only z differs.

  5. The z-axis is:

    • perpendicular to the xy-plane
    • in the xy-plane
    • parallel to x
    • the same as y
    Show answer and explanation

    Answer: perpendicular to the xy-plane

    Third dimension.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

Browse practice topics

Related lessons

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Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.