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Vector equation of a line

A straight line in 3D is described by a point on it and a direction. Every point on the line is the point plus some multiple of the direction.

The equation

r = a + λb, where a is the position vector of a point on the line, b is a direction vector and λ is a scalar. Always write it with 'r ='.

Through two points C and D: r = c + λ(d - c).

Points on a line

Each value of λ gives a point. To check whether a point lies on the line, find λ from one component and check that the same λ works in the others.

Parallel lines

Two lines are parallel if their direction vectors are multiples of each other. Their position vectors can be anything.

Worked example

Find a vector equation of the line through A(1, 2, 3) and B(4, 0, 5).

  1. Direction: b - a = (3, -2, 2)
  2. Use A as the point.

Answer: r = (i + 2j + 3k) + λ(3i - 2j + 2k)

Key idea: r = a + λb: a point plus a multiple of a direction. Check a point by finding one λ that works for every component.

Check yourself

  1. What is a direction vector of r = (2i - j + 4k) + λ(i + 3j - k)?
    • i + 3j - k
    • 2i - j + 4k
    • 3i + 2j + 3k
    • λ
    Show answer

    i + 3j - k. The direction vector is the one multiplied by λ.

  2. Which is a vector equation of the line through (1, 0, 2) and (3, 5, -1)?
    • r = i + 2k + λ(3i + 5j - k)
    • r = 2i + 5j - 3k + λ(i + 2k)
    • r = i + 2k + λ(2i + 5j - 3k)
    • r = i + 2k + λ(4i + 5j + k)
    Show answer

    r = i + 2k + λ(2i + 5j - 3k). Direction (3, 5, -1) - (1, 0, 2) = (2, 5, -3), through (1, 0, 2).

  3. The point (7, b, 7) lies on r = (i + 2j + 3k) + λ(3i - 2j + 2k). Find b.
    Show answer

    -2. From x: 1 + 3λ = 7, so λ = 2. Check z: 3 + 4 = 7. Then b = 2 - 4 = -2.

  4. Does (4, 0, 6) lie on r = (i + 2j + 3k) + λ(3i - 2j + 2k)?
    • Yes: λ = 1 fits every component
    • No: no value of λ gives x = 4
    • Yes: 4 + 0 + 6 = 1 + 2 + 3 + 4
    • No: λ = 1 fits x and y, but then z = 5
    Show answer

    No: λ = 1 fits x and y, but then z = 5. x: 1 + 3λ = 4 gives λ = 1. y: 2 - 2 = 0 works. z: 3 + 2 = 5, not 6.

  5. Find the x-coordinate where r = (i + 2j + 3k) + λ(3i - 2j + 2k) meets the plane z = 0.
    Show answer

    -3.5. z = 3 + 2λ = 0 gives λ = -1.5. Then x = 1 + 3(-1.5) = -3.5.

  6. Which line is parallel to r = (i + j) + λ(2i - j + 4k)?
    • r = (2i - j + 4k) + μ(i + j)
    • r = (i + j) + μ(2i + j + 4k)
    • r = μ(4i - j + 2k)
    • r = (3i + k) + μ(-6i + 3j - 12k)
    Show answer

    r = (3i + k) + μ(-6i + 3j - 12k). -6i + 3j - 12k = -3(2i - j + 4k).

  7. Find the distance from A(2, -1, 0) to the point with λ = 2 on r = (2i - j) + λ(3i + 4k).
    Show answer

    10. The movement is 2(3i + 4k) = 6i + 8k, whose length is sqrt(36 + 64) = 10.

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