Mathematics ยท WMA14 (P4)

Vectors in three dimensions

In P4 you work with vectors in 3D, written with i, j and k or as column vectors.

Notation

a = 2i - j + 3k can also be written as the column (2, -1, 3). i, j and k are unit vectors in the x, y and z directions.

Magnitude

The magnitude (length) of xi + yj + zk is sqrt(x2 + y2 + z2). A unit vector in the direction of a is a / |a|.

Between two points

The vector from A to B is AB = b - a, where a and b are the position vectors of A and B.

Worked example

A = (1, 2, 3) and B = (4, 6, 3). Find |AB|.

  1. AB = b - a = 3i + 4j + 0k
  2. |AB| = sqrt(9 + 16 + 0)

Answer: |AB| = 5

Key idea: |xi + yj + zk| = sqrt(x2 + y2 + z2); AB = b - a; unit vector = a/|a|.

Check yourself

  1. Find the magnitude of 2i + 3j + 6k.
    Show answer

    7. sqrt(4 + 9 + 36) = sqrt(49) = 7.

  2. Find |i - 2j + 2k|.
    Show answer

    3. sqrt(1 + 4 + 4) = 3.

  3. A has position vector i + j and B has position vector 3i + 5j + k. What is AB?
    • 4i + 6j + k
    • -2i - 4j - k
    • 2i + 4j + k
    • 3i + 5j
    Show answer

    2i + 4j + k. AB = b - a = (3-1)i + (5-1)j + (1-0)k.

  4. Which is a unit vector?
    • i + j
    • 0.6i + 0.8k
    • 2k
    • i + j + k
    Show answer

    0.6i + 0.8k. sqrt(0.36 + 0.64) = 1.

  5. Find the magnitude of 4i - 4j + 7k.
    Show answer

    9. sqrt(16 + 16 + 49) = sqrt(81) = 9.

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Sources

Last updated 7 November 2026. Spotted a mistake? Email alevel@lifereadysociety.com.