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11.6PureStretch

Modelling with Vectors

Vectors model anything with size and direction — velocity, displacement, force. This lesson applies the vector toolkit to real situations, where speed is a magnitude and direction is often a bearing.

25 min Vectors
Edexcel AS Level Maths: 11.6 Modelling with Vectors (Application)Curated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Model displacement and velocity as vectors
  • Find speed as the magnitude of velocity
  • Use vectors with bearings
  • Find resultant displacement and position over time
1

Velocity and speed

Velocity is a vector (it has direction); speed\textbf{speed} is its magnitude (just a number). So a velocity of (34)\begin{pmatrix} 3 \\ 4 \end{pmatrix} m/s corresponds to a speed of 55 m/s.

speed=∣velocity∣\text{speed} = |\text{velocity}|
Speed is the magnitude of the velocity vector.
Example 1A particle moves with velocity (68)\begin{pmatrix} 6 \\ 8 \end{pmatrix} m/s. Find its speed.
162+82=100\sqrt{6^2 + 8^2} = \sqrt{100}.
Answer1010 m/s
2

Position over time

If an object starts at position r0\mathbf{r}_0 and moves with constant velocity v\mathbf{v}, its position after time tt is found by adding tt lots of the velocity.

r=r0+t v\mathbf{r} = \mathbf{r}_0 + t\,\mathbf{v}
Start position plus (time × velocity).
Example 1Start at (12)\begin{pmatrix} 1 \\ 2 \end{pmatrix}, velocity (3−1)\begin{pmatrix} 3 \\ -1 \end{pmatrix}. Find the position after t=4t = 4.
1(12)+4(3−1)\begin{pmatrix} 1 \\ 2 \end{pmatrix} + 4\begin{pmatrix} 3 \\ -1 \end{pmatrix}.
Answer(13−2)\begin{pmatrix} 13 \\ -2 \end{pmatrix}
3

Vectors and bearings

In navigation problems the i\mathbf{i} direction is usually east and j\mathbf{j} north. Convert a bearing into a direction angle, sketch it, and use components to find resultant displacements.

Tip — Sketch east as i and north as j, then break each leg of the journey into components before adding.

Formula recap

speed=∣v∣\text{speed} = |\mathbf{v}|
Speed = magnitude of velocity.
r=r0+t v\mathbf{r} = \mathbf{r}_0 + t\,\mathbf{v}
Position under constant velocity.
i=east, j=north\mathbf{i} = \text{east}, \ \mathbf{j} = \text{north}
Usual navigation convention.

Common mistakes to avoid

Calling velocity and speed the same thing.
Velocity is a vector; speed is its magnitude (a scalar).
Adding time to the position instead of (time × velocity).
Position = r₀ + t·v — multiply the velocity by time first.

Key takeaways

  • Velocity is a vector; speed = |velocity|.
  • Position after time t: r = r₀ + t·v.
  • For bearings, take i = east, j = north and work in components.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Check your understanding

Work through questions on Modelling with Vectors. Use the feedback to decide what needs another look.

2 past-paper questions · 8 marks

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