All subjects · Mathematics · WME01 (M1)

Position vectors and constant velocity

When a particle moves with constant velocity, its position vector changes by the same amount every second. This gives the key formula r = r_0 + vt.

The formula

If a particle starts at position vector r_0 and moves with constant velocity v, then after t seconds

r = r_0 + vt.

Speed is the magnitude of the velocity: |v|.

Finding a velocity

Velocity = change in displacement ÷ time: v = (r_2 - r_1)/t. The direction of motion is the direction of v (find an angle or a bearing with tan).

Due north, due east

With i east and j north:

To find when it is due north of another point, set the i components equal.

Worked example

A particle starts at (1i + 6j) m and moves with velocity (2i - 3j) m/s. When is it due east of O?

  1. r = (1 + 2t)i + (6 - 3t)j
  2. Due east: 6 - 3t = 0, so t = 2
  3. Then the i part is 1 + 4 = 5 > 0, as needed

Answer: t = 2 s, at 5i m

Key idea: r = r_0 + vt; speed = |v|; v = (r_2 - r_1)/t. Due north: i part zero; due east: j part zero.

Check yourself

  1. A particle starts at (2i + 3j) m and moves with constant velocity (4i - j) m/s. Find its distance from O after 3 s.
    Show answer

    14 m. r = (2 + 12)i + (3 - 3)j = 14i, so it is 14 m from O.

  2. Find the speed of a particle moving with velocity (6i + 8j) m/s.
    Show answer

    10 m/s. sqrt(36 + 64) = 10 m/s.

  3. A particle moves with constant velocity from (i + 2j) m to (13i - 3j) m in 4 s. Find its speed, to 3 significant figures.
    Show answer

    3.25 m/s. v = (12i - 5j)/4 = 3i - 1.25j, so speed = sqrt(9 + 1.5625) = 3.25 m/s.

  4. A particle has position vector r = (1 + 2t)i + (6 - 3t)j metres at time t seconds. Find t when it is due east of O.
    Show answer

    2 s. Set the j component to zero: 6 - 3t = 0, so t = 2.

  5. A boat has position vector r = (-8 + 2t)i + (1 + t)j km at time t hours. Find its distance from O when it is due north of O.
    Show answer

    5 km. Due north: -8 + 2t = 0, so t = 4. Then r = 5j, so it is 5 km from O.

  6. With i east and j north, a ship moves with velocity (-5i + 5j) km/h. Find the bearing on which it is moving.
    Show answer

    315 °. It moves equally west and north, so its course is north-west, a bearing of 315°.

  7. A particle passes through the point with position vector a at t = 0 and has constant velocity b. Which gives its position at time t?
    • b + at
    • (a + b)t
    • a + bt
    • a - bt
    Show answer

    a + bt. Start at a and add bt, the displacement after t seconds.

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