12.4PureStretch

Application to Mechanics

Vectors describe physical quantities with both magnitude and direction — displacement, velocity, acceleration and force. Resolving and combining them solves mechanics problems in 3D.

22 min Video by Zeeshan Zamurred Vectors

Edexcel Pure Maths Year 2 — Chapter 12 Vectors (playlist)Watch the full walkthrough before the notes below.

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What you'll be able to do

Represent forces and velocities as vectors
Find resultant vectors
Find magnitude (speed) and direction
Apply vectors to motion problems

Vector quantities

Force, velocity and acceleration are vectors. The $resultant$ of several vectors is their sum, found by adding components. The magnitude of a velocity vector is the speed.

speed = ∣ v ∣ = v_{x}^{2} + v_{y}^{2} + v_{z}^{2}

Magnitude of a velocity vector.

∣ v ∣ = 3^{2} + 0^{2} + 4^{2}

= 25 = 5

Answer

5

m/s

Tip — Speed is the magnitude of velocity; resultant force is the vector sum of forces.

Resultants and equilibrium

Add force vectors component by component to get the resultant. A particle is in $equilibrium$ when the resultant force is the zero vector.

Formula recap

speed = ∣ v ∣

Magnitude of velocity.

F_{res} = \sum F_{i}

Resultant force.

\sum F = 0 \Rightarrow equilibrium

Balanced forces.

Common mistakes to avoid

Treating force or velocity as a scalar.

They are vectors — keep track of direction via components.

Adding magnitudes to get a resultant.

Add the vectors component by component, then take the magnitude.

Key takeaways

Force, velocity, acceleration are vectors.
Resultant = vector sum of components.
Speed = |v|; equilibrium ⇔ resultant force is zero.

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