Scalars and vectors

What you'll learn

How to tell whether a physical quantity is a scalar or a vector.
How to add and subtract vectors using arrows.
How to find the resultant of two coplanar vectors by scale drawing or calculation.
How to resolve a vector into perpendicular components using $Fx=Fcos⁡θF_x = F\cos\theta$ and $Fy=Fsin⁡θF_y = F\sin\theta$ .

Physical quantities: magnitude, unit and direction

A physical quantity is something measurable, such as mass, time, force or velocity. A complete measurement usually needs a magnitude and a unit.

For example, a mass might be 2.5 kg. The magnitude is 2.5 and the unit is kilograms, kg.

Some quantities also need a direction. A force of 10 N to the right is not the same as a force of 10 N to the left, even though the magnitudes are the same.

Definition

Magnitude

The magnitude of a quantity is its size, written as a positive value with a unit. For a vector, the magnitude is the length of the arrow representing it.

Definition

Scalar and vector quantities

A scalar quantity has magnitude only. A vector quantity has both magnitude and direction.

Examples of scalar quantities include:

mass, measured in kilograms, kg
time, measured in seconds, s
temperature, measured in kelvin, K
distance, measured in metres, m
speed, measured in metres per second, m s⁻¹
energy, measured in joules, J
power, measured in watts, W

Examples of vector quantities include:

displacement, measured in metres, m
velocity, measured in metres per second, m s⁻¹
acceleration, measured in metres per second squared, m s⁻²
force, measured in newtons, N
weight, measured in newtons, N
momentum, measured in kilogram metres per second, kg m s⁻¹

Key Idea

Direction matters

If changing the direction changes the physical effect, the quantity is a vector. If direction is irrelevant, the quantity is a scalar.

Distance and displacement

Distance is the total length of the path travelled. It is a scalar.

Displacement is the straight-line change in position from the start point to the finish point, including direction. It is a vector.

Common Mistake

Distance is not displacement

A journey can have a large distance but zero displacement. For example, if you run one complete lap of a circular track and return to the start, your distance is the circumference of the track, but your displacement is zero.

Example

Comparing distance and displacement

A student walks 30 m east, then 40 m north. Find the distance travelled and the displacement from the starting point.

Add the path lengths to find the scalar distance:
$m\text{distance} = 30\text{ m} + 40\text{ m} = 70\text{ m}$
Treat the east and north parts as perpendicular vector components, so the displacement magnitude is found using Pythagoras’ theorem:
$m\begin{aligned} s &= \sqrt{(30\text{ m})^2 + (40\text{ m})^2} \\ &= 50\text{ m} \end{aligned}$
Find the direction relative to east using trigonometry:
$tan⁡θ=4030θ=53∘\begin{aligned} \tan\theta &= \frac{40}{30} \\ \theta &= 53^\circ \end{aligned}$
The displacement is therefore 50 m at 53° north of east.

Representing vectors with arrows

A vector can be drawn as an arrow.

The length of the arrow represents the magnitude.
The arrowhead shows the direction.
The same vector can be moved around on the page as long as its length and direction do not change.

The magnitude of a vector $A⃗\vec{A}$ may be written as $∣A⃗∣|\vec{A}|$ , or simply $A$ when the context is clear.

Definition

Resultant vector

The resultant vector is the single vector that has the same overall effect as two or more vectors acting together.

Vector addition and subtraction

To add vectors, use the head-to-tail method:

Draw the first vector.
Draw the second vector starting from the head of the first.
Draw the resultant from the tail of the first vector to the head of the final vector.

The order of addition does not change the final resultant: $A⃗+B⃗=B⃗+A⃗\vec{A} + \vec{B} = \vec{B} + \vec{A}$ .

To subtract a vector, add the negative of that vector. The vector $−B⃗-\vec{B}$ has the same magnitude as $B⃗\vec{B}$ but points in the opposite direction.

A⃗−B⃗=A⃗+(−B⃗)\vec{A} - \vec{B} = \vec{A} + (-\vec{B})

The diagram below shows vector addition, subtraction, and the idea of resolving a vector into components.

Vector addition, subtraction and resolution diagrams

Common Mistake

Adding magnitudes only

You can only add vector magnitudes directly when the vectors act in exactly the same direction. If the directions differ, you must use a vector diagram or components.

Vector triangles

Two vectors are coplanar if they lie in the same plane. For A Level questions in this topic, that usually means they can be drawn on a flat page.

A vector triangle is formed when two vectors are placed head-to-tail and the resultant completes the triangle.

Scale drawing method

A scale drawing is a graphical method for finding a resultant.

Choose a sensible scale, such as 1 cm representing 2 N.
Draw the first vector accurately with a ruler and protractor.
Draw the second vector from the head of the first, at the correct angle.
Draw the resultant from the start point to the final point.
Measure the length and direction of the resultant, then convert the length using your scale.

Tip

Scale drawing accuracy

Use as large a diagram as the page allows. A tiny vector triangle makes percentage uncertainty in your measured length and angle much larger.

Calculation method

For a right-angled vector triangle, use Pythagoras’ theorem and basic trigonometry.

For non-right-angled triangles, you can use the cosine rule and sine rule, or resolve the vectors into perpendicular components. Components are usually the most reliable method in exam solutions.

Example

Finding a resultant force

Two coplanar forces act on an object. One force is 6.0 N east. The other is 8.0 N at 60° north of east. Find the magnitude and direction of the resultant force.

Resolve the 8.0 N force into east and north components:
$N\begin{aligned} F_{\text{east}} &= 8.0\cos 60^\circ = 4.0\text{ N} \\ F_{\text{north}} &= 8.0\sin 60^\circ = 6.93\text{ N} \end{aligned}$
Add the east components, because both forces have eastward parts:
$NF_{x} = 6.0\text{ N} + 4.0\text{ N} = 10.0\text{ N}$
The only northward component is from the 8.0 N force:
$NF_{y} = 6.93\text{ N}$
Use Pythagoras’ theorem to find the resultant magnitude:
$N\begin{aligned} R &= \sqrt{(10.0\text{ N})^2 + (6.93\text{ N})^2} \\ &= 12.2\text{ N} \end{aligned}$
Find the angle north of east:
$tan⁡θ=6.9310.0θ=34.7∘\begin{aligned} \tan\theta &= \frac{6.93}{10.0} \\ \theta &= 34.7^\circ \end{aligned}$
The resultant force is 12 N at 35° north of east, to two significant figures.

Resolving a vector into perpendicular components

To resolve a vector means to split it into components in chosen directions. The components are not extra forces or extra velocities; they are a mathematically equivalent way of describing the original vector.

Most commonly, vectors are resolved into horizontal and vertical components.

Definition

Components

The components of a vector are perpendicular parts which combine to give the original vector. For horizontal and vertical axes, these are often called the $x$ -component and $y$ -component.

If a force $F$ acts at an angle $θ\theta$ above the horizontal, then:

Fx=Fcos⁡θF_x = F\cos\theta

Fy=Fsin⁡θF_y = F\sin\theta

Here, $F_x$ is the horizontal component and $F_y$ is the vertical component.

Key Idea

Adjacent uses cosine

When the angle $θ\theta$ is measured from the horizontal, the horizontal component is adjacent to the angle, so $Fx=Fcos⁡θF_x = F\cos\theta$ . The vertical component is opposite the angle, so $Fy=Fsin⁡θF_y = F\sin\theta$ .

Common Mistake

Check where the angle is measured from

The equations $Fx=Fcos⁡θF_x = F\cos\theta$ and $Fy=Fsin⁡θF_y = F\sin\theta$ apply when $θ\theta$ is measured from the horizontal. If the angle is measured from the vertical, the sine and cosine swap roles.

Example

Resolving a force

A rope pulls a crate with a force of 45 N at 30° above the horizontal. Calculate the horizontal and vertical components of the force.

Identify that the angle is measured from the horizontal, so the horizontal component uses cosine:
$N\begin{aligned} F_x &= F\cos\theta \\ &= 45\text{ N}\cos 30^\circ \\ &= 39\text{ N} \end{aligned}$
Use sine for the vertical component because it is opposite the 30° angle:
$N\begin{aligned} F_y &= F\sin\theta \\ &= 45\text{ N}\sin 30^\circ \\ &= 23\text{ N} \end{aligned}$
Check that both components are smaller than the original 45 N force, which is sensible because each is only part of the original vector.

Choosing axes

You are free to choose axes that make the problem easier. Horizontal and vertical axes are common, but on a slope it is often better to choose axes parallel and perpendicular to the slope.

Once you choose positive directions, keep them consistent. Opposite directions should be treated as negative components.

Tip

Component method for awkward vectors

For several vectors in different directions, resolve every vector into $x$ and $y$ components, add all the $x$ components, add all the $y$ components, then recombine using Pythagoras and trigonometry.

Exam technique

In the exam

State whether a quantity is scalar or vector by referring to direction, not just by giving an example.
For vector diagrams, draw arrows head-to-tail and label the resultant from the start point to the final point.
Before using $Fx=Fcos⁡θF_x = F\cos\theta$ and $Fy=Fsin⁡θF_y = F\sin\theta$ , check that $θ\theta$ is measured from the horizontal.
Give resultant vectors with both magnitude and direction, including units such as N, m, or m s⁻¹.
For scale drawings, choose a clear scale and use a ruler and protractor carefully; for calculations, carry units through and round sensibly at the end.

Self review

Check yourself

What is the difference between speed and velocity?
Two forces of 5 N east and 12 N north act on an object. What is the resultant force?
A force of 20 N acts at 35° above the horizontal. Which component uses cosine, and why?

What you'll learn

How to tell whether a physical quantity is a scalar or a vector.
How to add and subtract vectors using arrows.
How to find the resultant of two coplanar vectors by scale drawing or calculation.
How to resolve a vector into perpendicular components using $Fx=Fcos⁡θF_x = F\cos\theta$ and $Fy=Fsin⁡θF_y = F\sin\theta$ .

Physical quantities: magnitude, unit and direction

A physical quantity is something measurable, such as mass, time, force or velocity. A complete measurement usually needs a magnitude and a unit.

For example, a mass might be 2.5 kg. The magnitude is 2.5 and the unit is kilograms, kg.

Some quantities also need a direction. A force of 10 N to the right is not the same as a force of 10 N to the left, even though the magnitudes are the same.

Definition

Magnitude

The magnitude of a quantity is its size, written as a positive value with a unit. For a vector, the magnitude is the length of the arrow representing it.