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 Video by Zeeshan Zamurred Vectors

Edexcel A-Level Maths: Vectors (Part 1)Watch the full walkthrough 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

The third axis

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

∣ O P ∣ = x^{2} + y^{2} + z^{2}

Distance from the origin.

2^{2} + 3^{2} + 6^{2} = 4 + 9 + 36

= 49 = 7

Answer

7

Distance between two points

The distance between $(x_{1}, y_{1}, z_{1})$ and $(x_{2}, y_{2}, z_{2})$ is $(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2}$ .

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

Formula recap

∣ O P ∣ = x^{2} + y^{2} + z^{2}

From the origin.

(Δ x)^{2} + (Δ y)^{2} + (Δ 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

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