Scalars & Vectors

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レッスンノート

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.
Example: A hiker's displacement is the straight-line distance from start to finish, while the distance walked is the total path length.
To decide if a quantity is a vector, ask: can it have a minus sign? For example, negative displacement is possible, but negative energy is not.

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 the horizontal component and F_y for the vertical component of force F.

Combining Vectors

Vectors can be combined by adding or subtracting them to produce the resultant vector (also called the 'net' vector).
Triangle method: 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.
Parallelogram method: Step 1: link the vectors tail-to-tail. Step 2: complete the parallelogram. Step 3: the resultant vector is the diagonal of the parallelogram.

Vector Multiplication

The product of a scalar and a vector is always a vector.
Example: mass (scalar) × acceleration (vector) = force (vector): F = m × a.
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 together have the same effect.
The two parts are called components.
For a force vector of magnitude F at an angle θ to the horizontal: horizontal component F_x = F cos θ, vertical component F_y = F sin θ.
The magnitude of the resultant vector is found using Pythagoras' Theorem.
The direction of the resultant vector is found from the angle it makes with the horizontal or vertical, using trigonometry (sine, cosine, or tangent).
Tip: cos θ is always the adjacent side of the right-angled triangle (making a 'cos sandwich').

Forces as Vectors

Forces are vectors and are often represented by free-body force diagrams.
Rules for drawing a free-body diagram: 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.
Common forces on objects: weight, normal reaction force, tension (from cords and strings), and friction.
Use conventional symbols: w or mg for weight; N or R for normal reaction force.

Vector diagram of two perpendicular forces resolved into a resultant force

Forces on an Inclined Plane

An inclined plane is a flat surface tilted at an angle θ.
The weight vector (W = mg) always acts vertically downwards.
On an inclined plane, weight can be split into components: perpendicular to the slope W = mg cos θ, parallel to the slope W = mg sin θ.
The normal (or reaction) force R is always perpendicular to the surface.
If there is no friction, the parallel component of weight, mg sin θ, causes the object to move down the slope.
If the object is not moving perpendicular to the slope, the normal force is R = mg cos θ.

Equilibrium

Forces are in equilibrium if an object is either at rest or moving at constant velocity.
In equilibrium, coplanar forces are represented by closed vector triangles.
The vectors, when joined together, form a closed path.
Three forces on an object in equilibrium form a closed vector triangle.

Systems in equilibrium

Scale Diagrams

Use calculation if the vectors are perpendicular; use a scale drawing if the vectors are not perpendicular.
Scale drawing involves drawing lengths and angles accurately using a sharp pencil, ruler, and protractor.
Steps: link vectors head-to-tail; draw the resultant using the triangle or parallelogram method; measure the length of the resultant with a ruler; measure the angle (from North if it is a bearing) with a protractor.
A scale may be given, such as 1 cm = 1 km.
The final answer must be converted back to the units needed, e.g. for a scale of 1 cm = 2 km, a 5 cm length represents 10 km.
Scale diagram questions typically involve vector triangles that do not contain a right angle.

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練習問題

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1.Which of the following is a scalar quantity?
Easy
- ADisplacement
- BVelocity
- CDistance
- DForce
2.Which of the following is a vector quantity?
Easy
- AMass
- BTime
- CDisplacement
- DSpeed
3.A scalar quantity has both magnitude and direction.
Easy
True or false?
4.Which of the following correctly describes the difference between distance and displacement?
Easy
- ADistance is a vector; displacement is a scalar.
- BDistance is a scalar; displacement is a vector.
- CBoth are scalars.
- DBoth are vectors.
5.A vector is represented by an arrow. What does the length of the arrow represent?
Easy
- AThe direction of the vector
- BThe magnitude of the vector
- CThe unit of the vector
- DThe component of the vector
6.Which of the following is NOT a scalar quantity?
Medium
- AEnergy
- BMass
- CVelocity
- DTemperature
7.The product of a scalar and a vector is always a vector.
Easy
True or false?
8.Which of the following gives the correct horizontal and vertical components of a force F at an angle θ to the horizontal?
Medium
- AHorizontal = F sin θ, vertical = F cos θ
- BHorizontal = F cos θ, vertical = F sin θ
- CHorizontal = F tan θ, vertical = F cos θ
- DHorizontal = F sin θ, vertical = F tan θ

もっと問題

9.Which of the following are vector quantities? (Select all that apply.)
Medium
- ADisplacement
- BSpeed
- CForce
- DEnergy
- EMomentum
10.A force of 250 kN acts at 15° to the vertical. What is the horizontal component of this force?
Medium
- A242 kN
- B64.7 kN
- C250 kN
- D15 kN
11.An object is in equilibrium on an inclined plane at angle θ to the horizontal. Which expression gives the component of its weight parallel to the slope?
Hard
- Amg cos θ
- Bmg sin θ
- Cmg tan θ
- Dmg
12.When resolving a vector, the magnitude of the resultant vector can be found using Pythagoras' theorem.
Easy
True or false?
13.In a scale diagram, a scale of 1 cm = 2 km is used. A resultant vector is measured as 5 cm on the diagram. What is the actual magnitude of the vector?
Medium
- A2.5 km
- B5 km
- C10 km
- D20 km
14.Place the steps for finding a resultant vector using a scale drawing in the correct order.
Medium
- Measure the length of the resultant vector using a ruler.
- Link the vectors head-to-tail if they aren't already.
- Measure the angle of the resultant vector using a protractor.
- Draw the resultant vector using the triangle or parallelogram method.
15.Which of the following is NOT one of the rules for drawing a free-body diagram?
Medium
- ADraw a point at the centre of mass of the body.
- BDraw the body free from contact with any other object.
- CDraw the forces acting on the body using vectors with length proportional to magnitude.
- DDraw the forces acting on other objects in the same diagram.
16.Which of the following statements about vectors are correct? (Select all that apply.)
Medium
- AA vector has both magnitude and direction.
- BThe resultant vector is also known as the net vector.
- CVectors can only be combined by addition, not subtraction.
- DThe parallelogram method involves linking vectors tail-to-tail.
- EScalars can be resolved into components.
17.Match each quantity to whether it is a scalar or a vector.
Medium
- Distance
- Displacement
- Speed
- Velocity
- Scalar
- Vector
18.Which of the following pairs contains one scalar quantity and one vector quantity?
Easy
- ADistance and displacement
- BSpeed and distance
- CVelocity and acceleration
- DMass and energy
19.Which of the following statements about vector multiplication are correct? (Select all that apply.)
Medium
- AThe product of a scalar and a vector is always a vector.
- BForce is the product of mass (scalar) and acceleration (vector).
- CMomentum is the product of mass (scalar) and velocity (vector).
- DThe product of two scalars is always a vector.
- EThe product of a scalar and a vector is always a scalar.
20.A quantity that can have a negative value is likely to be a vector quantity.
Easy
True or false?
21.Match each vector combination method to its correct first step.
Medium
- Triangle method
- Parallelogram method
- Scale drawing
- Link the vectors head-to-tail
- Link the vectors tail-to-tail
- Choose a sensible scale
22.Place the steps for using a scale drawing to find a resultant vector in the correct order.
Medium
- Choose a sensible scale
- Draw the vectors head-to-tail
- Draw the resultant vector using the triangle or parallelogram method
- Measure the length of the resultant vector with a ruler
- Measure the angle of the resultant vector with a protractor
23.An object on a rough inclined plane at angle θ to the horizontal is in equilibrium. What is the normal reaction force R in terms of the object's weight W?
Medium
- AR = W cos θ
- BR = W sin θ
- CR = W tan θ
- DR = W
24.Which of the following are conditions for forces to be in equilibrium? (Select all that apply.)
Medium
- AThe object is at rest.
- BThe object is moving at constant velocity.
- CThe vector triangle of the forces is closed.
- DThe object is accelerating.
- EThe resultant force is zero.
25.A scale diagram uses a scale of 1 cm = 2 km. A resultant vector is measured as 5 cm on the diagram. What is the actual magnitude of the vector?
Medium
- A10 km
- B2.5 km
- C5 km
- D20 km
26.A boat is pulled by two ropes, each exerting a force of 500 N, with an angle of 60° between the ropes. Which method is most appropriate to determine the magnitude of the resultant force?
Medium
- AScale drawing
- BPythagoras' theorem
- CSimple trigonometry using sine and cosine
- DVector addition by calculation using the cosine rule
27.A scalar quantity is multiplied by a vector quantity. What is the nature of the resulting quantity?
Medium
- AAlways a vector
- BAlways a scalar
- CCould be either a scalar or a vector
- DIt depends on the units
28.A force of 300 N acts at an angle of 40° above the horizontal. What is the vertical component of this force?
Medium
- A193 N
- B230 N
- C300 N
- D252 N
29.An object of weight 50 N is on a smooth inclined plane at 25° to the horizontal. What is the component of its weight parallel to the slope?
Medium
- A21 N
- B45 N
- C50 N
- D23 N
30.A vector of magnitude 10 units is resolved into two perpendicular components. One component has magnitude 6 units. What is the magnitude of the other component?
Medium
- A4 units
- B8 units
- C16 units
- D2 units
31.A quantity that can have a negative value is always a vector quantity.
Easy
True or false?
32.Match each quantity with its correct classification.
Medium
- Distance
- Displacement
- Speed
- Velocity
- Scalar
- Vector
33.What is the magnitude of the hiker's resultant displacement?
Medium
- A11.7 km
- B16 km
- C8 km
- D12.5 km
34.A hiker walks a distance of 6 km due east and 10 km due north. What is the magnitude of their displacement?
Medium
- A11.7 km
- B16 km
- C8 km
- D4 km
35.A hiker walks a distance of 6 km due east and 10 km due north. What is the direction of their displacement from the horizontal?
Medium
- A59°
- B31°
- C45°
- D37°
36.Which method is most appropriate for combining two vectors that are not perpendicular?
Medium
- AScale drawing
- BCalculation using Pythagoras' theorem
- CCalculation using trigonometry
- DVector multiplication
37.A hiker walks 6 km due east and then 10 km due north. What is the magnitude of their displacement, to 2 significant figures?
Medium
- A11.66 km
- B12 km
- C16 km
- D8 km
38.A hiker walks 6 km due east and then 10 km due north. What is the direction of their displacement from the horizontal?
Medium
- A31°
- B59°
- C45°
- D90°
39.A helicopter provides a lift of 250 kN when the blades are tilted at 15° from the vertical. What is the vertical component of the lift force?
Hard
- A64.7 kN
- B242 kN
- C250 kN
- D259 kN
40.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?
Hard
- A64.7 kN
- B242 kN
- C250 kN
- D259 kN
41.Complete the sentence about vector addition using the triangle method.
Easy
To combine vectors using the triangle method, link the vectors ____ to ____.
42.A hiker walks 6 km due east and then 10 km due north. Calculate the magnitude of the resultant displacement in km, giving your answer to 2 significant figures.
Medium
43.A force of 250 kN acts at 15° to the vertical. Calculate the horizontal component of this force in kN, giving your answer to 3 significant figures.
Medium
44.Complete the sentence about resolving vectors.
Easy
The horizontal component of a force F at angle θ to the horizontal is F cos θ, and the vertical component is F ____ θ.
45.What is the name of the single vector that has the same effect as two or more vectors combined?
Medium
46.A force of 200 N acts at an angle of 30° to the horizontal. Calculate the horizontal component of the force in N, giving your answer to 3 significant figures.
Medium
47.An object of weight 80 N rests on a smooth inclined plane at 20° to the horizontal. Calculate the normal reaction force in N, giving your answer to 3 significant figures.
Medium
48.Complete the sentence about vector multiplication.
Medium
The product of a scalar and a vector is always a ____.
49.Calculate the direction of the resultant displacement from the horizontal, in degrees, giving your answer to the nearest degree.
Medium
50.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 in kN, giving your answer to 3 significant figures.
Medium
51.A helicopter provides a lift of 250 kN when the blades are tilted at 15° from the vertical. Calculate the horizontal component of the lift force in kN, giving your answer to 3 significant figures.
Medium
52.Complete the sentences about resolving vectors.
Medium
For a force F at an angle θ to the horizontal, the horizontal component is F ____ θ and the vertical component is F ____ θ.
53.Explain why displacement is a vector quantity but distance is a scalar quantity.
Medium
54.A weight hangs in equilibrium from two cables. State the condition for equilibrium and describe how the forces can be represented.
Hard
55.Explain why the weight of an object on an inclined plane can be resolved into components parallel and perpendicular to the slope.
Hard
56.A hiker walks 6 km due east and then 10 km due north. Calculate the magnitude of the resultant displacement in km, giving your answer to 2 significant figures.
Medium
- A13 km
- B14 km
- C11 km
- D12 km
57.A force of 250 kN acts at 15° to the vertical. Calculate the horizontal component of this force in kN, giving your answer to 3 significant figures.
Medium
- A58.2 kN
- B64.7 kN
- C77.7 kN
- D71.2 kN
58.A force of 200 N acts at an angle of 30° to the horizontal. Calculate the horizontal component of the force in N, giving your answer to 3 significant figures.
Medium
- A190 N
- B207 N
- C173 N
- D156 N
59.An object of weight 80 N rests on a smooth inclined plane at 20° to the horizontal. Calculate the normal reaction force in N, giving your answer to 3 significant figures.
Medium
- A82.7 N
- B75.2 N
- C67.7 N
- D90.2 N
60.Calculate the direction of the resultant displacement from the horizontal, in degrees, giving your answer to the nearest degree.
Medium
- A53 °
- B59 °
- C65 °
- D71 °
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 in kN, giving your answer to 3 significant figures.
Medium
- A266 kN
- B242 kN
- C218 kN
- D290 kN
62.A helicopter provides a lift of 250 kN when the blades are tilted at 15° from the vertical. Calculate the horizontal component of the lift force in kN, giving your answer to 3 significant figures.
Medium
- A64.7 kN
- B71.2 kN
- C58.2 kN
- D77.7 kN

フラッシュカード

Define a scalar quantity.

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

Define 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: can this quantity have a minus sign? If yes, it is a vector; if no, it is a scalar. For example, negative displacement is possible, but negative energy is not.

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 a resultant vector?

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

Describe the triangle method for combining two 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 two 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); mass × velocity (vector) = momentum (vector).

What does it mean to resolve a vector?

Resolving a vector means representing a single resultant vector by two vectors (components) which, in combination, have the same effect as the original one. It is the opposite of adding vectors.

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

Use Pythagoras' Theorem: if the components are x and y, the resultant magnitude is √(x² + 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 calculate the direction of a resultant vector from its components?

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

What are the rules for drawing a free-body force 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 weight parallel and perpendicular to the slope?

Parallel to the slope: W = mg sin θ. Perpendicular to the slope: W = mg cos θ. Weight always acts vertically downwards.

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, what is the normal force if the object is not moving perpendicular to the slope?

The normal force is R = mg cos θ, acting perpendicular to the surface.

What are the conditions for forces to be 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 by vectors?

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

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 vector using the triangle or parallelogram method. Step 3: Measure the length of the resultant using a ruler. Step 4: Measure the angle of the resultant (from North if it is a bearing) using 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.

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