All subjects · Mathematics · WMA14 (P4)

Vector geometry and distance in 3D

Vectors describe movements and positions in 2D and 3D. You use them to show lines are parallel, to divide a line in a ratio and to find distances.

Adding and scaling

a + b is a followed by b. ka is parallel to a and k times as long (opposite direction if k < 0). Two vectors are parallel if one is a scalar multiple of the other: 2i - 3j + k is parallel to -4i + 6j - 2k.

Collinear points and ratios

A, B and C are collinear if AB is a multiple of BC (they share point B).

If P divides AB in the ratio m : n, then p = a + (m/(m + n))(b - a).

Distance between two points

The distance between (x_1, y_1, z_1) and (x_2, y_2, z_2) is sqrt((x_2 - x_1)2 + (y_2 - y_1)2 + (z_2 - z_1)2). This is the magnitude of the vector between them.

Worked example

A(1, 2, -1) and B(4, -1, 5). Find AB and the distance AB.

  1. AB = b - a = 3i - 3j + 6k
  2. |AB| = sqrt(9 + 9 + 36) = sqrt(54)

Answer: sqrt(54) = 3sqrt(6) ≈ 7.35

Key idea: Parallel means a scalar multiple. Collinear means parallel through a shared point. Distance = sqrt(Δx2 + Δy2 + Δz2).

Check yourself

  1. Find the distance between (2, -1, 3) and (5, 3, 15).
    Show answer

    13. Differences 3, 4 and 12. sqrt(9 + 16 + 144) = sqrt(169) = 13.

  2. Which vector is parallel to 2i - 3j + k?
    • 4i - 6j - 2k
    • 2i + 3j + k
    • -4i + 6j - 2k
    • i - 3j + 2k
    Show answer

    -4i + 6j - 2k. -4i + 6j - 2k = -2(2i - 3j + k).

  3. The vector 3i + pj - 6k is parallel to i + 2j - 2k. Find p.
    Show answer

    6. The i-components show the multiplier is 3 (and -6 = 3 × -2 confirms it). So p = 3 × 2 = 6.

  4. A(1, 4, 2) and B(7, 1, 8). P lies on AB with AP : PB = 1 : 2. Find P.
    • (5, 2, 6)
    • (4, 2.5, 5)
    • (2, 3.5, 3)
    • (3, 3, 4)
    Show answer

    (3, 3, 4). p = a + (1/3)(b - a) = (1, 4, 2) + (1/3)(6, -3, 6) = (3, 3, 4).

  5. How can you show that points A, B and C are collinear?
    • Show that |AB| = |BC|
    • Show that AB = k BC for some scalar k
    • Show that AB · BC = 0
    • Show that a + b = c
    Show answer

    Show that AB = k BC for some scalar k. Parallel vectors through the shared point B lie on one line.

  6. a = 2i + j - k and b = i - 3j + 2k. Find |2a - b|, to 2 decimal places.
    Show answer

    7.07. 2a - b = 3i + 5j - 4k, so |2a - b| = sqrt(9 + 25 + 16) = sqrt(50) = 7.07.

  7. A(1, 0, 2) and B(3, t, 5), with t > 0. Given |AB| = 7, find t.
    Show answer

    6. AB = (2, t, 3), so 4 + t2 + 9 = 49. t2 = 36 and t = 6.

Practise with instant feedback

Sources

Last updated 7 November 2026. Spotted a mistake? Tell us.