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M10.2MechanicsCore

Forces as Vectors

Forces are vectors, so several forces combine into a single resultant by vector addition. Working in i–j components turns force problems into straightforward arithmetic.

25 min Forces and Motion
Edexcel AS Level Maths: 10.2 Forces as VectorsCurated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Express forces in i–j form
  • Find the resultant of several forces
  • Find the magnitude and direction of the resultant
  • Apply equilibrium in component form
1

Resultant force

The resultant\textbf{resultant} of several forces is their vector sum. In i–j form, add the i\mathbf{i} components and the j\mathbf{j} components separately.

Fres=∑F\mathbf{F}_{\text{res}} = \sum \mathbf{F}
Add forces componentwise.
Example 1Find the resultant of (3i+2j)(3\mathbf{i} + 2\mathbf{j}) N and (i−5j)(\mathbf{i} - 5\mathbf{j}) N.
1Add components: (3+1)i+(2−5)j(3+1)\mathbf{i} + (2-5)\mathbf{j}.
Answer4i−3j4\mathbf{i} - 3\mathbf{j} N
2

Magnitude and direction

The size of the resultant force is its magnitude (Pythagoras), and its direction is found with tan⁡−1\tan^{-1} of the components — just like any vector.

∣F∣=a2+b2|\mathbf{F}| = \sqrt{a^2 + b^2}
e.g. ∣4i−3j∣=5|4\mathbf{i} - 3\mathbf{j}| = 5 N.
3

Equilibrium in components

An object is in equilibrium when the resultant is zero, which means the i\mathbf{i} components sum to zero and\textbf{and} the j\mathbf{j} components sum to zero. This gives two equations to solve for unknown forces.

Tip — Equilibrium ⟶ set the total i-component = 0 and the total j-component = 0 separately.

Formula recap

Fres=∑F\mathbf{F}_{\text{res}} = \sum \mathbf{F}
Resultant = vector sum.
∣F∣=a2+b2|\mathbf{F}| = \sqrt{a^2 + b^2}
Magnitude of resultant.
equilibrium⇒∑Fi=0, ∑Fj=0\text{equilibrium} \Rightarrow \sum F_i = 0, \ \sum F_j = 0
Component equilibrium.

Common mistakes to avoid

Adding magnitudes of forces instead of components.
Forces are vectors — add their i and j components separately.
Using one equation for 2D equilibrium.
Equilibrium gives two equations: i-components = 0 and j-components = 0.

Key takeaways

  • Resultant force = vector sum (add components).
  • Magnitude via Pythagoras; direction via tan⁻¹.
  • Equilibrium: total i-component = 0 and total j-component = 0.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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