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 Video by Zeeshan Zamurred Forces and Motion

Edexcel AS Level Maths: 10.2 Forces as VectorsWatch the full walkthrough 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

Resultant force

The $resultant$ of several forces is their vector sum. In i–j form, add the $i$ components and the $j$ components separately.

F_{res} = \sum F

Add forces componentwise.

1Add components:

(3 + 1) i + (2 - 5) j

Answer

4 i - 3 j

Magnitude and direction

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

∣ F ∣ = a^{2} + b^{2}

e.g.

∣4 i - 3 j ∣ = 5

Equilibrium in components

An object is in equilibrium when the resultant is zero, which means the $i$ components sum to zero $and$ the $j$ 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

F_{res} = \sum F

Resultant = vector sum.

∣ F ∣ = a^{2} + b^{2}

Magnitude of resultant.

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.

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