Vectors in Three Dimensions

This lesson extends vectors to three dimensions using the i, j, k notation — as required by the Edexcel A-Level Mathematics specification (9MA0). You need to perform vector operations in 3D, calculate magnitudes, and find unit vectors in three dimensions.

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Extending to 3D

In three dimensions, we add a z-component to our vectors. A 3D vector is written as:

a = (a₁, a₂, a₃) or equivalently a₁i + a₂j + a₃k

where:

• i = (1, 0, 0) — unit vector along the x-axis
• j = (0, 1, 0) — unit vector along the y-axis
• k = (0, 0, 1) — unit vector along the z-axis

Example: v = (3, -1, 5) = 3i - j + 5k

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Magnitude in 3D

The magnitude of a = (a₁, a₂, a₃) is:

|a| = √(a₁² + a₂² + a₃²)

This is the 3D extension of Pythagoras' theorem.

Example 1: |a| where a = (2, 3, 6) |a| = √(4 + 9 + 36) = √49 = 7

Example 2: |b| where b = (1, -2, 2) |b| = √(1 + 4 + 4) = √9 = 3