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 Video by Zeeshan Zamurred Vectors

Edexcel AS Level Maths: 11.6 Modelling with Vectors (Application)Watch the full walkthrough 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

Velocity and speed

Velocity is a vector (it has direction); $speed$ is its magnitude (just a number). So a velocity of $(34)$ m/s corresponds to a speed of $5$ m/s.

speed = ∣ velocity ∣

Speed is the magnitude of the velocity vector.

6^{2} + 8^{2} = 100

Answer

10

m/s

Position over time

If an object starts at position $r_{0}$ and moves with constant velocity $v$ , its position after time $t$ is found by adding $t$ lots of the velocity.

r = r_{0} + t v

Start position plus (time × velocity).

(12) + 4 (3 - 1)

Answer

(13 - 2)

Vectors and bearings

In navigation problems the $i$ direction is usually east and $j$ 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 ∣

Speed = magnitude of velocity.

r = r_{0} + t v

Position under constant velocity.

i = east, j = 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.

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