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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 Vectors
Edexcel Pure Maths Year 2 — Chapter 12 Vectors (playlist)Curated by Kepler, not produced by us. Watch it 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
1

Vector quantities

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

speed=∣v∣=vx2+vy2+vz2\text{speed} = |\mathbf{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}
Magnitude of a velocity vector.
Example 1A particle has velocity v=3i+4k\mathbf{v} = 3\mathbf{i} + 4\mathbf{k} m/s. Find its speed.
1∣v∣=32+02+42|\mathbf{v}| = \sqrt{3^2 + 0^2 + 4^2}.
2=25=5= \sqrt{25} = 5.
Answer
55 m/s

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

2

Resultants and equilibrium

Add force vectors component by component to get the resultant. A particle is in equilibrium\textbf{equilibrium} when the resultant force is the zero vector.

Formula recap

speed=∣v∣\text{speed} = |\mathbf{v}|
Magnitude of velocity.
Fres=∑Fi\mathbf{F}_{\text{res}} = \sum \mathbf{F}_i
Resultant force.
∑F=0⇒equilibrium\sum \mathbf{F} = \mathbf{0} \Rightarrow \text{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.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. Which is a vector quantity?

    • velocity
    • speed
    • mass
    • time
    Show answer and explanation

    Answer: velocity

    Has direction.

  2. The speed of 3i+4k3\mathbf{i} + 4\mathbf{k} m/s is:

    • 55
    • 77
    • 7\sqrt{7}
    • 1212
    Show answer and explanation

    Answer: 55

    √(9+16).

  3. The resultant of forces is found by:

    • adding components
    • adding magnitudes
    • multiplying
    • subtracting speeds
    Show answer and explanation

    Answer: adding components

    Vector sum.

  4. A particle is in equilibrium when the resultant force is:

    • the zero vector
    • maximum
    • a unit vector
    • horizontal
    Show answer and explanation

    Answer: the zero vector

    Forces balance.

  5. Speed is the ____ of velocity.

    • magnitude
    • direction
    • square
    • reciprocal
    Show answer and explanation

    Answer: magnitude

    |v|.

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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Related lessons

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Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.