M8.4MechanicsCore

Working with Vectors

Mechanics quantities like velocity and force are vectors, written in i–j form. Adding them, and finding magnitude and direction, are exactly the vector skills from Pure — now applied to motion and forces.

25 min Video by Zeeshan Zamurred Modelling in Mechanics

Edexcel Mechanics Year 1 — full playlist (Zeeshan Zamurred)Watch the full walkthrough before the notes below.

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What you'll be able to do

Write mechanics vectors in i–j form
Add and subtract vectors in components
Find the magnitude (e.g. speed) of a vector
Find the direction of a vector

i–j form in mechanics

Forces, velocities and displacements are written using $i$ (horizontal) and $j$ (vertical) unit vectors, e.g. a velocity $v = 3 i + 4 j$ m/s. Combine them by adding components, just as in Pure.

Magnitude

The $magnitude$ of a vector is found with Pythagoras. For a velocity vector, the magnitude is the $speed$ ; for a force, it is the size of the force.

∣ a i + b j ∣ = a^{2} + b^{2}

Magnitude — e.g. speed from a velocity vector.

3^{2} + 4^{2} = 25

Answer

5

m/s

Direction

The direction is the angle the vector makes, found with $tan^{- 1}$ of the components — typically measured from the positive $i$ direction. A sketch ensures you get the correct angle.

θ = tan^{- 1} (\frac{b}{a})

Direction of

a i + b j

Tip — Magnitude of a velocity vector is the speed; the direction is the bearing/angle of travel.

Formula recap

v = a i + b j

Vector in i–j form.

∣ v ∣ = a^{2} + b^{2}

Magnitude (e.g. speed).

θ = tan^{- 1} (b / a)

Direction.

Common mistakes to avoid

Adding i and j components together.

Keep components separate: add i to i and j to j.

Confusing speed with velocity.

Speed is the magnitude of the velocity vector.

Key takeaways

Mechanics vectors are written in i–j form and added componentwise.
Magnitude = √(a² + b²); for velocity this is the speed.
Direction = tan⁻¹(b/a) — sketch to place the angle.

Test yourself

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