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Subtracting vectors

AshPhys · Revision cheat sheet
AS Level9702 AS Ch.1Chapter 1: KinematicsLesson 1.7Sections 1.7

Goal: Find a change in velocity, Δv = v₂ − v₁, by adding the reverse of the first vector.

velocityacceleration

01The big picture

To subtract a vector, add its reverse: a⃗−b⃗=a⃗+(−b⃗)\vec{a} - \vec{b} = \vec{a} + (-\vec{b}), where −b⃗-\vec{b} has the same size as b⃗\vec{b} but points the opposite way.

The most common use is a change in velocity: Δv⃗=v⃗2−v⃗1\Delta\vec{v} = \vec{v}_2 - \vec{v}_1 (final minus initial). Even at constant speed, a change of direction gives a non-zero Δv\Delta v.

Change in velocity [paraphrase]
Final velocity minus initial velocity, found as a vector.
Fig. 1 · Δv = v₂ − v₁ = v₂ + (−v₁)
Δv = v₂ − v₁ = v₂ + (−v₁)Left: the initial velocity v1 pointing north and the final velocity v2 pointing east, equal in length. Right: v2 drawn east, then minus v1 drawn south from its tip, and delta v from the start of v2 to the end of minus v1, pointing south-east.v1v2v2−v1Δv
Draw v₂, then add the reverse of v₁ at its tip. Δv runs from the start of v₂ to the end of −v₁.

02Key equations

Subtracting a vector
a⃗−b⃗=a⃗+(−b⃗)\vec{a} - \vec{b} = \vec{a} + (-\vec{b})
Change in velocity
Δv⃗=v⃗2−v⃗1\Delta\vec{v} = \vec{v}_{2} - \vec{v}_{1}
Symbols and units
  • v⃗1, v⃗2\vec{v}_{1},\ \vec{v}_{2}initial and final velocitym s−1\mathrm{m\,s^{-1}}
  • Δv⃗\Delta\vec{v}change in velocitym s−1\mathrm{m\,s^{-1}}

03Sample problems with solutions

Sample 1 · A car turning a corner
A car travelling north at 20 m s⁻¹ turns and travels east at 20 m s⁻¹. Find its change in velocity.

  1. Add the reverse of v1v_1 (20 m s⁻¹ south) to v2v_2 (20 m s⁻¹ east): the two are perpendicular.

    ∣Δv∣=202+202=28.3 m s−1|\Delta v| = \sqrt{20^{2} + 20^{2}} = 28.3\ \mathrm{m\,s^{-1}}

  2. Direction: equal east and south parts, so 45° between them.

Answer: 28 m s⁻¹ towards the south-east (45° south of east)

04Practice problems

  1. P1
    A ball moving right at 6.0 m s⁻¹ rebounds moving left at 4.0 m s⁻¹. Find the change in velocity.
    Final answer10 m s⁻¹ to the left
    Answer: 10 m s⁻¹ to the left
  2. P2
    A velocity changes from 3.0 m s⁻¹ east to 4.0 m s⁻¹ north. Find Δv.
    Final answer5.0 m s⁻¹ at 53° north of west
    Answer: 5.0 m s⁻¹ at 53° north of west
  3. P3
    Find A − B for A = 7.0 N east and B = 2.0 N east.
    Final answer5.0 N east
    Answer: 5.0 N east
  4. P4
    A velocity changes from 10 m s⁻¹ north to 10 m s⁻¹ south. Find Δv.
    Final answer20 m s⁻¹ south
    Answer: 20 m s⁻¹ south

05Other useful information

Common mistakes

  • ✗ Wrong: Subtract the speeds: 20 − 20 = 0, so no change.

    ✓ Right: The direction changed, so Δv is not zero.

  • ✗ Wrong: Work out v₁ − v₂.

    ✓ Right: Δv = final − initial: v₂ − v₁.

Exam tips

  • In one dimension, give each velocity a sign (+ one way, − the other) and subtract the numbers.

What the syllabus asks

  • 9702 topic 1: add and subtract coplanar vectors. Coursebook 1.7.

Explore: Practice questions for this lesson