Scalars & Vectors

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Lesson notes

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.

Slides

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Practice questions

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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

Flashcards

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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