Scalars & Vectors

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Scalar & Vector Quantities

A scalar is a quantity that has only a magnitude (size).
A vector is a quantity that has both a magnitude and a direction.
Distance is a scalar because it describes how far an object has travelled overall, but not the direction.
Displacement is a vector because it describes how far an object is from its starting point and in what direction.
For example, a hiker's displacement is the straight-line distance from start to finish, while the distance walked is the total path length.

Representing Vectors

Vectors are represented by an arrow.
The arrowhead indicates the direction of the vector.
The length of the arrow represents the magnitude of the vector.
Component vectors are sometimes drawn with a dotted line and a subscript indicating horizontal or vertical (e.g., F_x for horizontal component, F_y for vertical component).

Examiner Tips and Tricks

To decide if a quantity is a vector or scalar, ask: can it have a minus sign?
If a negative value is possible (e.g., negative displacement), it is a vector.
If a negative value is not possible (e.g., negative energy), it is a scalar.

Combining Vectors

Vectors can be combined by adding or subtracting them to produce the resultant vector.
The resultant vector is sometimes called the 'net' vector (e.g., net force).
Triangle method: link vectors head-to-tail; the resultant is formed by connecting the tail of the first vector to the head of the second.
Parallelogram method: link vectors tail-to-tail; complete the parallelogram; the resultant is the diagonal.

Vector Multiplication

The product of a scalar and a vector is always a vector.
For example, mass (scalar) × acceleration (vector) = force (vector): F = m × a.
Another example: mass (scalar) × velocity (vector) = momentum (vector): p = m × v.

Resolving Vectors

Resolving a vector is the opposite of adding vectors: a single resultant vector is represented by two vectors that have the same combined effect.
The magnitude of the resultant vector can be found using Pythagoras' Theorem if the components are perpendicular.
When a vector is broken down, the parts are called components.
For a force vector of magnitude F at an angle θ to the horizontal, the horizontal component is F_x = F cos θ and the vertical component is F_y = F sin θ.
The direction of the resultant vector is found from the angle it makes with the horizontal or vertical, using trigonometry (sine, cosine, or tangent).

Force as a Vector

Vectors are used in many areas of physics, especially motion, forces, and momentum.
Forces are often represented by free-body force diagrams.
Rules for free-body diagrams: draw a point at the centre of mass; draw the body free from contact; draw forces as vectors with length proportional to magnitude; draw the tail from the centre of mass and use the tip to indicate direction.
On an inclined plane, the weight vector (W = mg) can be split into components parallel and perpendicular to the slope.
Perpendicular component of weight: W = mg cos θ; parallel component: W = mg sin θ.
The normal (reaction) force R is perpendicular to the surface; if the object is not moving perpendicular to the slope, R = mg cos θ.
If there is no friction, the parallel component mg sin θ causes the object to move down the slope.

Vector diagram of two perpendicular forces resolved into a resultant force

Equilibrium

Forces are in equilibrium if an object is at rest or moving at constant velocity.
In equilibrium, coplanar forces can be represented by a closed vector triangle.
The vectors, when joined together, form a closed path.
Common forces on objects include weight, normal reaction force, tension, and friction.
When labelling force vectors, use conventional symbols such as w or mg for weight, and N or R for normal reaction force.

Systems in equilibrium

Scale Diagrams

Vectors can be combined or resolved by calculation (if perpendicular) or by scale drawing (if not perpendicular).
Scale drawing involves accurately drawing lengths and angles using a sharp pencil, ruler, and protractor.
Steps for scale drawing: link vectors head-to-tail; draw the resultant using triangle or parallelogram method; measure the length with a ruler; measure the angle with a protractor.
A scale may be given (e.g., 1 cm = 1 km); convert measured lengths back to real units.
Scale diagram questions typically involve vector triangles that do not contain a right angle.

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1.Which of the following is the correct definition of a vector quantity?
Easy
- AA quantity that has both magnitude and direction
- BA quantity that has only magnitude
- CA quantity that has only direction
- DA quantity that has magnitude, direction, and a unit
2.Which of the following is a scalar quantity?
Easy
- ADisplacement
- BVelocity
- CDistance
- DForce
3.Displacement is a vector quantity.
Easy
True or false?
4.A hiker walks a distance of 6 km due east and then 10 km due north. What is the magnitude of the hiker's displacement?
Easy
- A4 km
- B8 km
- C11.7 km
- D16 km
5.A hiker walks 6 km due east and then 10 km due north. What is the direction of the hiker's displacement from the horizontal (east direction)?
Easy
- A31°
- B45°
- C59°
- D72°
6.A helicopter provides a lift of 250 kN when the blades are tilted at 15° from the vertical. What is the horizontal component of the lift force?
Medium
- A64.7 kN
- B242 kN
- C250 kN
- D15 kN
7.Which of the following quantities are vectors? (Select all that apply.)
Medium
- ADisplacement
- BDistance
- CVelocity
- DSpeed
- EForce
8.Match each quantity to its correct type (scalar or vector).
Medium
- Mass
- Velocity
- Energy
- Force
- Scalar
- Vector

Plus de questions

9.Place the steps for combining two vectors using the triangle method in the correct order.
Medium
- Link the vectors head-to-tail
- Draw the resultant vector from the tail of the first vector to the head of the second vector
- Measure the length and angle of the resultant if required
10.An object is in equilibrium under the action of three coplanar forces. Which statement about the vector triangle representing these forces is correct?
Medium
- AThe vectors form an open triangle
- BThe vectors form a closed triangle
- CThe vectors are all perpendicular
- DThe vectors all point in the same direction
11.The product of a scalar and a vector is always a vector.
Easy
True or false?
12.Which of the following quantities are scalars? (Select all that apply.)
Medium
- Adistance
- Bdisplacement
- Cspeed
- Dvelocity
- Emass
13.Match each vector quantity to its scalar counterpart.
Medium
- displacement
- velocity
- force
- speed
- mass
- distance
14.Place the steps for combining two vectors using the parallelogram method in the correct order.
Medium
- Link the vectors tail-to-tail
- Complete the parallelogram
- Draw the resultant vector as the diagonal
15.Which of the following statements about resolving vectors are correct? (Select all that apply.)
Medium
- AResolving a vector is the opposite of adding vectors.
- BA single vector can be replaced by two perpendicular components that have the same overall effect.
- CThe horizontal component of a force F at angle θ to the horizontal is F sin θ.
- DThe vertical component of a force F at angle θ to the horizontal is F sin θ.
- EThe magnitude of the resultant of two perpendicular components can be found using Pythagoras' theorem.
16.An object is in equilibrium. Which of the following must be true?
Medium
- AThe object is at rest.
- BThe object is moving at constant velocity.
- CThe object is either at rest or moving at constant velocity.
- DThe net force on the object is zero.
17.An object of weight W is on a smooth inclined plane that makes an angle θ with the horizontal. What is the component of weight parallel to the slope?
Medium
- AW cos θ
- BW sin θ
- CW tan θ
- DW / sin θ
18.An object of mass 5.0 kg is on a smooth inclined plane that makes an angle of 30° with the horizontal. What is the component of the object's weight parallel to the slope? (Take g = 9.8 m/s².)
Medium
- A24.5 N
- B42.4 N
- C49.0 N
- D9.8 N
19.An object of mass 5.0 kg is on a smooth inclined plane that makes an angle of 30° with the horizontal. What is the normal reaction force acting on the object? (Take g = 9.8 m/s².)
Medium
- A42.4 N
- B24.5 N
- C49.0 N
- D9.8 N
20.A box is at rest on a rough inclined plane. Which of the following statements are correct? (Select all that apply.)
Hard
- AThe normal reaction force is equal to mg cos θ.
- BThe frictional force is equal to mg sin θ.
- CThe weight of the box is mg.
- DThe net force on the box is zero.
- EThe normal reaction force is equal to mg sin θ.
21.A hiker walks 6 km due east and then 10 km due north. Which of the following gives the magnitude and direction of the hiker's displacement?
Medium
- A11.7 km at 59° north of east
- B16.0 km at 59° north of east
- C11.7 km at 31° north of east
- D11.7 km at 59° east of north
22.Which of the following is a vector quantity?
Medium
- AMomentum
- BMass
- CEnergy
- DTemperature
23.A force of 100 N acts at an angle of 60° to the horizontal. What is the vertical component of the force?
Medium
- A86.6 N
- B50.0 N
- C100 N
- D0 N
24.A force of 100 N acts at an angle of 60° to the horizontal. What is the horizontal component of the force?
Medium
- A50.0 N
- B86.6 N
- C100 N
- D0 N
25.Which of the following is the correct expression for the component of a force F parallel to a slope inclined at an angle θ to the horizontal?
Medium
- AF sin θ
- BF cos θ
- CF tan θ
- DF / sin θ
26.An object of weight W rests on a smooth inclined plane at an angle θ to the horizontal. Which expression gives the normal reaction force R?
Medium
- AW cos θ
- BW sin θ
- CW tan θ
- DW
27.A force of 60 N acts at an angle of 35° above the horizontal. What is the horizontal component of the force?
Medium
- A49 N
- B34 N
- C60 N
- D73 N
28.A vector has a horizontal component of 8.0 N and a vertical component of 6.0 N. What is the magnitude of the vector?
Medium
- A10 N
- B14 N
- C2.0 N
- D48 N
29.Which of the following statements about vector multiplication is correct?
Medium
- AThe product of a scalar and a vector is a vector
- BThe product of a scalar and a vector is a scalar
- CThe product of two vectors is always a scalar
- DThe product of two scalars is a vector
30.Which of the following statements about an object in equilibrium are correct? (Select all that apply.)
Medium
- AThe object is at rest or moving at constant velocity
- BThe vector sum of the forces on the object is zero
- CThe forces on the object form a closed vector triangle if they are coplanar
- DThe object must be accelerating
- EThe object must have zero mass
31.A scalar quantity has magnitude only.
Easy
True or false?
32.Distance is a vector quantity.
Easy
True or false?
33.For an object on a smooth inclined plane, the component of weight parallel to the slope is mg cos θ.
Medium
True or false?
34.Match each quantity to its correct type.
Medium
- Velocity
- Speed
- Acceleration
- Distance
- Force
- Vector
- Scalar
35.Place the steps for determining a resultant vector using a scale drawing in the correct order.
Medium
- Choose a sensible scale
- Draw the vectors accurately using a ruler and protractor
- Draw the resultant vector using the triangle or parallelogram method
- Measure the length of the resultant and convert using the scale
- Measure the angle of the resultant using a protractor
36.When should a scale drawing be used instead of calculation to combine two vectors?
Medium
- AWhen the vectors are perpendicular
- BWhen the vectors are not perpendicular
- CWhen the vectors are parallel
- DWhen the vectors are equal in magnitude
37.Which of the following are required to produce an accurate scale drawing of vector addition? (Select all that apply.)
Medium
- AA sharp pencil
- BA ruler
- CA protractor
- DA calculator
- EA set square
38.In a scale drawing, the length of the arrow representing a vector is proportional to the magnitude of the vector.
Easy
True or false?
39.Match each step in creating a scale diagram with its correct description.
Medium
- Step 1
- Step 2
- Step 3
- Step 4
- Link the vectors head-to-tail
- Draw the resultant vector using the triangle or parallelogram method
- Measure the length of the resultant vector using a ruler
- Measure the angle of the resultant vector using a protractor
40.Which scale would be most appropriate for the scale drawing?
Medium
- A1 cm = 0.1 N
- B1 cm = 1 N
- C1 cm = 10 N
- D1 cm = 100 N
41.A force of 50 N acts at an angle of 30° to the horizontal. Calculate the horizontal component of the force. Give your answer to 3 significant figures.
Medium
42.Complete the sentence about resolving vectors.
Easy
The horizontal component of a force F at angle θ to the horizontal is given by F____x = F ____, and the vertical component is F____y = F ____.
43.A force of 80 N acts at an angle of 40° to the horizontal. Calculate the horizontal component of the force. Give your answer to 3 significant figures.
Medium
44.What is the name given to the two perpendicular vectors that combine to give a single resultant vector?
Medium
45.An object is suspended from a stationary rope. The tension in the rope is 120 N. Calculate the weight of the object.
Medium
46.A helicopter provides a lift of 250 kN when the blades are tilted at 15° from the vertical. Calculate the vertical component of the lift force. Give your answer in kN to 3 significant figures.
Medium
47.A force of 25 N acts at an angle of 30° to the horizontal. Calculate the horizontal component of the force. Give your answer to 2 significant figures.
Medium
48.A force of 40 N acts at an angle of 60° to the horizontal. Calculate the vertical component of the force. Give your answer to 2 significant figures.
Medium
49.A hiker walks 3 km due east, then 4 km due north. Calculate the magnitude of the hiker's displacement.
Medium
50.A hiker walks 3 km due east, then 4 km due north. Calculate the direction of the hiker's displacement measured from the horizontal (east direction). Give your answer to 2 significant figures.
Hard
51.An object of mass 2.0 kg is on a smooth inclined plane that makes an angle of 20° with the horizontal. Calculate the component of the object's weight parallel to the slope. Use g = 9.8 m/s² and give your answer to 2 significant figures.
Medium
52.An object of mass 2.0 kg is on a smooth inclined plane that makes an angle of 20° with the horizontal. Calculate the normal reaction force acting on the object. Use g = 9.8 m/s² and give your answer to 2 significant figures.
Medium
53.Explain the difference between distance and displacement, and state whether each is a scalar or a vector.
Hard
54.Explain why displacement is a vector quantity while distance is a scalar quantity.
Hard
55.A weight hangs in equilibrium from a cable at point X. The tensions in the cables are T₁ and T₂ as shown. Describe how you would draw a free-body diagram to represent the forces acting at point X.
Hard
56.State what is meant by a vector quantity and give one example.
Medium
57.Describe the difference between distance and displacement.
Medium
58.A force of 50 N acts at an angle of 30° to the horizontal. Calculate the horizontal component of the force. Give your answer to 3 significant figures.
Medium
- A47.6 N
- B43.3 N
- C39 N
- D51.9 N
59.A force of 80 N acts at an angle of 40° to the horizontal. Calculate the horizontal component of the force. Give your answer to 3 significant figures.
Medium
- A67.4 N
- B73.5 N
- C61.3 N
- D55.2 N
60.An object is suspended from a stationary rope. The tension in the rope is 120 N. Calculate the weight of the object.
Medium
- A120 N
- B108 N
- C144 N
- D132 N
61.A helicopter provides a lift of 250 kN when the blades are tilted at 15° from the vertical. Calculate the vertical component of the lift force. Give your answer in kN to 3 significant figures.
Medium
- A290 kN
- B218 kN
- C242 kN
- D266 kN
62.A force of 25 N acts at an angle of 30° to the horizontal. Calculate the horizontal component of the force. Give your answer to 2 significant figures.
Medium
- A22 N
- B24 N
- C26 N
- D20 N
63.A force of 40 N acts at an angle of 60° to the horizontal. Calculate the vertical component of the force. Give your answer to 2 significant figures.
Medium
- A43 N
- B35 N
- C31 N
- D39 N
64.A hiker walks 3 km due east, then 4 km due north. Calculate the magnitude of the hiker's displacement.
Medium
- A7 km
- B5 km
- C4 km
- D6 km
65.A hiker walks 3 km due east, then 4 km due north. Calculate the direction of the hiker's displacement measured from the horizontal (east direction). Give your answer to 2 significant figures.
Hard
- A63 °
- B53 °
- C58 °
- D48 °
66.An object of mass 2.0 kg is on a smooth inclined plane that makes an angle of 20° with the horizontal. Calculate the component of the object's weight parallel to the slope. Use g = 9.8 m/s² and give your answer to 2 significant figures.
Medium
- A6.7 N
- B7.7 N
- C8.7 N
- D5.7 N
67.An object of mass 2.0 kg is on a smooth inclined plane that makes an angle of 20° with the horizontal. Calculate the normal reaction force acting on the object. Use g = 9.8 m/s² and give your answer to 2 significant figures.
Medium
- A20 N
- B17 N
- C18 N
- D19 N

Cartes mémo

What is a scalar quantity?

A scalar is a quantity which only has a magnitude (size).

What is a vector quantity?

A vector is a quantity which has both a magnitude and a direction.

Give an example of a scalar quantity and explain why it is scalar.

Distance is a scalar because it describes how an object has travelled overall, but not the direction it has travelled in.

Give an example of a vector quantity and explain why it is a vector.

Displacement is a vector because it describes how far an object is from where it started and in what direction.

How can you quickly decide if a quantity is a vector or a scalar?

Ask yourself: can this quantity have a minus sign? For example, negative displacement is possible, so displacement is a vector; negative energy is not possible, so energy is a scalar.

How are vectors represented in diagrams?

Vectors are represented by an arrow: the arrowhead indicates the direction, and the length of the arrow represents the magnitude.

What is the resultant vector?

The resultant vector is the single vector that results from combining two or more vectors; it is sometimes called the 'net' vector (e.g., net force).

Describe the triangle method for combining vectors.

Step 1: link the vectors head-to-tail. Step 2: the resultant vector is formed by connecting the tail of the first vector to the head of the second vector.

Describe the parallelogram method for combining vectors.

Step 1: link the vectors tail-to-tail. Step 2: complete the resulting parallelogram. Step 3: the resultant vector is the diagonal of the parallelogram.

What is the result of multiplying a scalar by a vector?

The product of a scalar and a vector is always a vector. For example, mass (scalar) × acceleration (vector) = force (vector).

What does it mean to resolve a vector?

Resolving a vector means breaking it down into two perpendicular components that together have the same effect as the original vector.

How do you find the magnitude of a resultant vector from its perpendicular components?

Use Pythagoras' Theorem: magnitude = √(component_x² + component_y²).

For a force vector of magnitude F at an angle θ to the horizontal, what are its horizontal and vertical components?

Horizontal component: F_x = F cos θ. Vertical component: F_y = F sin θ.

How do you find the direction of a resultant vector from its components?

Use trigonometry: tan θ = opposite/adjacent, where θ is the angle from the horizontal or vertical, then θ = tan⁻¹(opposite/adjacent).

What are the four rules for drawing a free-body diagram?

Rule 1: Draw a point at the centre of mass. Rule 2: Draw the body free from contact with any other object. Rule 3: Draw forces as vectors with length proportional to magnitude. Rule 4: Draw the tail of each vector from the centre of mass and use the tip to indicate direction.

On an inclined plane at angle θ, what are the components of the weight W = mg parallel and perpendicular to the slope?

Perpendicular to the slope: W = mg cos θ. Parallel to the slope: W = mg sin θ.

On an inclined plane with no friction, what causes the object to move down the slope?

The parallel component of weight, mg sin θ, causes the object to move down the slope.

On an inclined plane, if the object is not moving perpendicular to the slope, what is the normal force R?

The normal force is R = mg cos θ.

When are forces in equilibrium?

Forces are in equilibrium if an object is either at rest or moving at constant velocity.

How are coplanar forces in equilibrium represented?

In equilibrium, coplanar forces are represented by closed vector triangles: the vectors, when joined together, form a closed path.

What are the most common forces acting on objects?

Weight, normal reaction force, tension (from cords and strings), and friction.

When should you use calculation versus scale drawing to combine or resolve vectors?

Use calculation if the vectors are perpendicular; use scale drawing if the vectors are not perpendicular.

What are the steps for determining a resultant vector using a scale drawing?

Step 1: Link the vectors head-to-tail. Step 2: Draw the resultant using the triangle or parallelogram method. Step 3: Measure the length of the resultant with a ruler. Step 4: Measure the angle of the resultant (from North if it is a bearing) with a protractor.

How do you convert a measured length on a scale drawing to the actual magnitude?

Use the given scale. For example, for a scale of 1 cm = 2 km, a resultant vector with a length of 5 cm measured on your ruler is actually 10 km.

What is the 'cos sandwich' tip for resolving vectors?

If unsure which component is cos θ or sin θ, remember that cos θ is always the adjacent side of the right-angled triangle.

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