The shape of the lesson
By the end of the lesson, learners can
- Tip to tail, then split: Set two vectors and watch their resultant form nose to tail, with its perpendicular components drawn in.
- Which is which: Knowing the category is worth easy marks.
- Subtracting is adding the reverse.
- Add nose to tail: the resultant runs start of first to end of last.
- Resolve: component along the axis the angle is from uses cosine.
- Subtract: reverse the second vector, then add.
One picture every idea returns to
A scalar has magnitude only; a vector has magnitude and direction. Coplanar vectors add nose to tail, the resultant running from the start of the first to the end of the last. Any single vector can be split into two perpendicular components, of size F cosθ along one axis and F sinθ along the other, which together carry exactly the same effect as the original.
Forty-five minutes, phase by phase
| Time | Phase | What happens in the room | Grouping |
|---|---|---|---|
| 0 to 5 min | Hook: predict | Open with the question in the title, Some quantities point somewhere.. Mass, energy and temperature are fully described by a number and a unit. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 13 min | Teach and check: Tip to tail, then split | Set two vectors and watch their resultant form nose to tail, with its perpendicular components drawn in. Project the simulation and let a student drive it while the class predicts. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Which is which | Knowing the category is worth easy marks. | Pairs |
| 21 to 30 min | Teach and check: Subtracting is adding the reverse | Subtracting is adding the reverse. | Cooperative Concept Mapping |
| 30 to 36 min | Check | The four Stage 1 questions from the student lesson, answered on mini whiteboards or by show of fingers. Re-teach the one most get wrong. | Individual, then whole class |
| 36 to 43 min | Plenary | The exam-style questions from Stage 2, one at a time: thirty seconds alone, then a random call. The feedback lines in the key give the examiner's wording. | Individual, random call |
| 43 to 45 min | Exit and homework | Set the worksheet below (6 questions, 14 marks). Learners self-assess against the objectives before they leave. | Individual |
Protect the parts that carry the learning
The sequence carries about fifty minutes of material, so the lab is driven by one student while the class predicts rather than explored freely. Times are a guide.
Protected: the check at 30 minutes. It is the only individual read on whether the key idea landed, and it is not cut.
Cut first if the build overruns: the last check in Subtracting is adding the reverse, never the check or the plenary.
In a 60 minute block: run worksheet questions 1 to 3 in class as a Rally Coach set and model both the hardest worked example and its check live.
Cooperative Concept Mapping
Teams receive the key terms on cards and arrange them on a large sheet, drawing and labelling the links. Compare maps between teams before showing the model map.
The full structure, with its variations and the other twelve, is on the cooperative-learning page.
What to head off, and how
| Trap learners fall into | Teaching move that pre-empts it |
|---|---|
| Do not add vectors by adding their magnitudes. | Two 5 N forces give 10 N only when they point the same way; at right angles the resultant is about 7.1 N, and head-on they cancel to zero. The second classic slip is the cos/sin mix-up when resolving: the component along the axis the angle is measured from uses cosine, and the perpendicular one uses sine. Always sketch the triangle and check which side is adjacent to the angle. |
Scalars and vectors: worksheet
6 questions, 14 marks. Written in the style of the papers; no past paper question is reproduced.
State the difference between a scalar and a vector quantity, and classify each of these: distance, velocity, mass, force.
Two forces, 6.0 N due east and 8.0 N due north, act on a point object. Find the magnitude and direction of the resultant force.
A crate of weight 200 N sits on a ramp inclined at 25° to the horizontal. Resolve its weight into components along and perpendicular to the ramp surface.
A ball moving at 5.0 m s⁻¹ to the right strikes a wall and rebounds at 5.0 m s⁻¹ to the left. The speed is unchanged, yet there is a change in velocity. Find the magnitude of the change in velocity and explain why it is not zero.
Explain, with a sketch in words, why three coplanar forces acting on an object in equilibrium can be represented by a closed triangle.
A student claims that adding a 4 N force and a 3 N force always gives 7 N. State when this is true and give the smallest possible magnitude of the resultant of these two forces.
Answer key and marking notes
- A scalar has magnitude only; a vector has magnitude and direction. ✓
- Scalars: distance, mass. Vectors: velocity, force. ✓
- Magnitude: R = √(6.0² + 8.0²) = √100 = 10 N. ✓
- Direction: tanθ = 8.0 / 6.0, so θ = 53° north of east. ✓
- Resultant: 10 N at 53° north of east. ✓
- Component along the ramp (down the slope): 200 sin 25° ≈ 85 N. ✓
- Component perpendicular to the ramp (into the surface): 200 cos 25° ≈ 181 N. ✓
- Take right as positive: u = +5.0, v = −5.0 m s⁻¹. ✓
- Δv = v − u = −5.0 − (+5.0) = −10 m s⁻¹, magnitude 10 m s⁻¹ (directed left). ✓
- Velocity is a vector; the direction reversed, so even with equal speeds the velocity changed by 10 m s⁻¹. ✓
- In equilibrium the resultant force is zero. ✓
- Drawing the three forces nose to tail, a zero resultant means the path returns to its starting point, so the three vectors form a closed triangle. ✓
- The sum 7 N is the resultant only when the two forces point in the same direction (parallel). ✓
- The smallest magnitude is when they are opposite: 4 − 3 = 1 N. The resultant can be anywhere from 1 N to 7 N depending on the angle. ✓
18 multiple-choice questions on this lesson
From the topic question bank. Assign them in the app for a marked score in your gradebook, or run the topic as a class game.
Which of these is a scalar quantity?
- AForce
- BVelocity
- CAcceleration
- DSpeed
Which of these is a vector quantity?
- AAcceleration
- BEnergy
- CDistance
- DTime
Which pair contains two vectors?
- AMass and energy
- BTime and temperature
- CDisplacement and force
- DSpeed and distance
Two forces of 5.0 N and 12 N act at right angles. The magnitude of the resultant is:
- A7 N
- B13 N
- C60 N
- D17 N
Two perpendicular forces of 8.0 N and 6.0 N give a resultant of magnitude:
- A48 N
- B2 N
- C14 N
- D10 N
A force of 100 N acts at 60° above the horizontal. Its vertical component is about:
- A87 N
- B58 N
- C100 N
- D50 N
A force of 100 N acts at 60° above the horizontal. Its horizontal component is:
- A58 N
- B100 N
- C50 N
- D87 N
A box of weight 60 N rests on a slope at 30° to the horizontal. The component of weight down the slope is:
- A52 N
- B30 N
- C35 N
- D60 N
Two equal forces of 10 N each act with an angle of 120° between them. The magnitude of their resultant is:
- A20 N
- B17 N
- C0 N
- D10 N
A 7 N force and a 4 N force act on a point. The largest possible magnitude of their resultant is:
- A11 N
- B3 N
- C7 N
- D28 N
A particle moving at 3.0 m s⁻¹ due east later moves at 3.0 m s⁻¹ due west. The magnitude of the change in velocity is:
- A3.0 m s⁻¹
- B9.0 m s⁻¹
- C6.0 m s⁻¹
- D0 m s⁻¹
To resolve a force F acting at angle θ to the x-axis, the y-component is:
- AF cosθ
- BF sinθ
- CF
- DF tanθ
If the vector triangle drawn for three forces on an object does not close, the object:
- Ais moving at constant velocity
- Bis in equilibrium
- Chas zero acceleration
- Dhas a non-zero resultant force
A boat heads due north at 4.0 m s⁻¹ across a river flowing due east at 3.0 m s⁻¹. The boat's resultant speed is:
- A5.0 m s⁻¹
- B1.0 m s⁻¹
- C3.5 m s⁻¹
- D7.0 m s⁻¹
A force of 50 N acts at 37° to the x-axis (with cos 37° = 0.80 and sin 37° = 0.60). Its x and y components are:
- A30 N and 40 N
- B50 N and 50 N
- C40 N and 30 N
- D25 N and 25 N
Forces of 5.0 N due east and 12 N due north act on a point. The resultant makes an angle to the east direction of about:
- A45°
- B67°
- C23°
- D53°
Two forces of 6.0 N act at right angles to each other. The magnitude of the resultant is:
- A12 N
- B0 N
- C3.0 N
- D8.5 N
Two scalar quantities of 4 units and 7 units are added. The result is:
- A11 units, regardless of any directions
- B13 units
- C3 units
- Dbetween 3 and 11 units depending on direction
Quiz key
Speed is a scalar (magnitude only). Velocity, force and acceleration are all vectors.
Acceleration is a vector. Distance, energy and time are scalars.
Displacement and force are both vectors. The other pairs are all scalars.
R = √(5² + 12²) = √169 = 13 N.
R = √(8² + 6²) = √100 = 10 N.
Vertical component = 100 sin 60° ≈ 87 N.
Horizontal component = 100 cos 60° = 50 N.
Component along the slope = 60 sin 30° = 60 × 0.5 = 30 N.
Each force contributes 10 cos 60° = 5 N along the bisector, giving 5 + 5 = 10 N.
When parallel they add directly: 7 + 4 = 11 N. The minimum, when opposite, is 3 N.
Taking east as positive: Δv = (−3.0) − (+3.0) = −6.0 m s⁻¹, magnitude 6.0 m s⁻¹.
The component perpendicular to the axis the angle is measured from uses sine: F sinθ.
A triangle that fails to close means the vectors do not sum to zero, so there is a resultant force and the object is not in equilibrium.
Resultant speed = √(4² + 3²) = √25 = 5.0 m s⁻¹.
x = 50 cos 37° = 40 N; y = 50 sin 37° = 30 N.
tanθ = 12 / 5.0, so θ = 67° measured from the east direction (north of east).
R = √(6² + 6²) = 6√2 ≈ 8.5 N.
Scalars simply add arithmetically: 4 + 7 = 11. Only vectors give a range that depends on the angle between them.
What to project
The student lesson embeds these tabs. Open them full screen on the projector; every slider has a drag handle on the stage.
Slides for this lesson
A projectable deck for every pack is on the way. Until then, present the student lesson full screen: its sections, figure and lab tabs follow the same order as the plan above.