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F = ma with vectors

Forces given in i, j form can be added like any vectors. Newton's second law then works component by component: F = ma, where F and a are vectors.

Resultant force

Add all the force vectors to get the resultant F. A particle is in equilibrium when the resultant is 0i + 0j.

F = ma as vectors

a = F/m, so the acceleration has the same direction as the resultant force. Divide each component by the mass. Then use vector suvat: v = u + at.

Unknown components

If you know the acceleration (or that a force is parallel to a given vector), compare i parts and j parts separately to form two equations.

Worked example

Forces (2i + 3j) N, (pi - 5j) N and (4i + qj) N act on a 3 kg particle, giving acceleration (4i + j) m/s2. Find p and q.

  1. Resultant = (6 + p)i + (q - 2)j.
  2. ma = 12i + 3j.
  3. 6 + p = 12 and q - 2 = 3.

Answer: p = 6, q = 5

Key idea: Add the forces, then F = ma component by component. The acceleration points the same way as the resultant force.

Check yourself

  1. A force (6i - 8j) N acts on a particle of mass 2 kg. Find the magnitude of its acceleration.
    Show answer

    5 m/s2. a = 3i - 4j, so |a| = 5 m/s2.

  2. A particle of mass 0.5 kg starts from rest and is acted on by a constant force (1.5i - 2j) N. Find its speed after 4 s.
    Show answer

    20 m/s. a = 3i - 4j. v = 4(3i - 4j) = 12i - 16j, so speed = 20 m/s.

  3. Forces (3i + 2j) N and (pi - 7j) N act on a 2 kg particle and its acceleration is (4i - 2.5j) m/s2. Find p.
    Show answer

    5 N. i parts: 3 + p = 2 × 4 = 8, so p = 5.

  4. A resultant force (5i + 5j) N acts on a particle. Find the angle between its acceleration and the vector i.
    Show answer

    45 °. The acceleration is parallel to the force, which makes 45° with i.

  5. A particle is in equilibrium under forces (4i - 3j) N, (-6i + 7j) N and F. Find |F|, to 3 significant figures.
    Show answer

    4.47 N. F = -(-2i + 4j) = 2i - 4j, so |F| = sqrt(20) = 4.47 N.

  6. Forces (2i + j) N and (ki + 3j) N act on a particle. Their resultant is parallel to i + 2j. Find k.
    Show answer

    0. Resultant = (2 + k)i + 4j. Parallel to i + 2j: 4 = 2(2 + k), so k = 0.

  7. What is true about the acceleration produced by a resultant force F on a particle of mass m?
    • It is mF, in the same direction as F
    • It is F/m, in the same direction as F
    • It is F/m, perpendicular to F
    • It depends on the particle's velocity
    Show answer

    It is F/m, in the same direction as F. a = F/m: dividing a vector by a positive number keeps its direction.

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