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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 Modelling in Mechanics
Edexcel Mechanics Year 1 — full playlist (Zeeshan Zamurred)Curated by Kepler, not produced by us. Watch it 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
1

i–j form in mechanics

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

2

Magnitude

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

∣ai+bj∣=a2+b2|a\mathbf{i} + b\mathbf{j}| = \sqrt{a^2 + b^2}
Magnitude — e.g. speed from a velocity vector.
Example 1A particle has velocity 3i+4j3\mathbf{i} + 4\mathbf{j} m/s. Find its speed.
132+42=25\sqrt{3^2 + 4^2} = \sqrt{25}.
Answer55 m/s
3

Direction

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

θ=tan⁡−1 ⁣(ba)\theta = \tan^{-1}\!\left(\dfrac{b}{a}\right)
Direction of ai+bja\mathbf{i} + b\mathbf{j}.

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

Formula recap

v=ai+bj\mathbf{v} = a\mathbf{i} + b\mathbf{j}
Vector in i–j form.
∣v∣=a2+b2|\mathbf{v}| = \sqrt{a^2 + b^2}
Magnitude (e.g. speed).
θ=tan⁡−1(b/a)\theta = \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

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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