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Bridging Course - Lecture 3 (Vectors and Scalars)

In this lecture the following are introduced:
• Descartes and Cartesian co-ordinates
• Vectors and scalars
• Polar co-ordinates
• Adding vectors sequentially and simultaneously
• Subtracting vectors
• Vector components and adding by components

Where am I?

This question has either many answers or none; because it is incomplete. In spatial terms it probably asks: "where am I, with reference to my home, or my destination?"

Cartesian Coordinates
Descartes portrait

Rene Descartes (1596 to 1650) said, we need a reference point and reference direction and from this we can uniquely identify every point in a plane (or in space).

He continued with this approach and created what is known as Coordinate Geometry.

The Cartesian co-ordinate system has an origin and two (or three) perpendicular axes, "x" and "y" (and "z"). A point is space is specified by co-ordinates along each axis.

cartesion coordinates

Distance and displacement

Once we can uniquely define where we are, we have to distinguish between distance and displacement.

The distance you measure between two points depends on the actual path you take between the points (and there are an infinite number of paths you can take!).

distance graphic

The displacement has the length and direction of the minimum straight-line between the two points.

displacement graphic

Vectors and Scalars

Because displacement has both length and direction, two numbers are needed to describe it. It is called a "vector" quantity. Distance only needs one number. Quantities that are described by one number are called "scalars". Examples of scalars are mass, angle and time.
To distinguish between vectors and scalars, textbooks use boldface type for vectors and plainface for scalars.

You need to distinguish between displacement (written as r) and distance (written as r).

In Cartesian co-ordinates:
a displacement is described as r = (rx ,  ry).
It is also written with vector displacements as r = rx + ry
where rx is a vector along the "x" (or say, Easterly) axis,
and ry is a vector along the "y" (or say, Northerly) axis.

vector representation

A vector of length one unit (a "unit vector") along the x-axis is written as i and a unit vector along the y-axis is written as j. Hence rx = rxi  and  ry = ryj.

So r = rx  +  ry  =  rxi   +  ryj = (rx ,  ry)

unit vector representation

Polar co-ordinates

There is another system called the Polar co-ordinate system, where you specify a displacement by its straight-line length and its angle from the reference direction.

In the Polar co-ordinate system,
r = (r , θ), where the way θ is defined depends on the angle convention and circular measure.

polar coordinates

Example: an angle which is 22O towards the East of South.

Angle conventions


measured as

angle is:


clockwise from North



smaller angle to cardinal point

South 22O East


anti-clockwise from positive x axis


Circular measure

1 revolution =



400 grads


1024 brads


64,000 mils


2π (Greek: 2*pi) radians


Angles measured in radians are preferred.

The angle between two vectors

This can be tricky at times, so it is best to put the two "tails" of the arrows together and then measure the smaller angle.

angle between vectors

Changing between Cartesian and Polar

r = (400m, 300m) in Cartesian notation is shown in the diagram below.


For Polar notation with maths angle convention.



so, r = (500m, 370)

Adding displacements

Displacements add in a different way from familiar quantities like mass, angle and time. One way to show how they add is to draw a scale diagram.

Here are two displacements, 3000 km Sydney to Darwin then 2600 km Darwin to Perth. They are equal to a 3200 km displacement Sydney to Perth.

adding displacements

This triangle diagram is clear when the displacements are sequential. In the triangle method you place the tail of the second arrow on the head of the first arrow and then join start to finish.

When things happen simultaneously, a parallelogram diagram is appropriate (though in practice, either can be used for either situation).

adding displacements

Subtracting vectors

To subtract vectors you simply add the negative. The negative of a vector has the same size but is in the opposite direction to its positive direction.

subtracting vectors

Formal definition of Vector quantities

Vectors are quantities which have
• size (or magnitude),
• direction, and
• add like displacements.


A force is a physical push or pull. It is vector quantity because both size and direction matter. The size is measured in a unit called the Newton. When a force is seen to have effects at right angles, the parts of the force in those directions are called "components".

This can be seen in the case of a yacht where the wind force (at right angle to the sail area) pushes the yacht forward but also tilts it sideways.


We can express this in a diagram this as follows

component diagram

Component Notation

If North is vertically up the page, then θ is an azimuthal angle and in polar coordinates, the force F = (F , θ). It would be written in Cartesian notation (with x as the East direction) as (F·Sinθ , F·Cosθ)

But note carefully:
If the maths convention is to be used, then the angle would be the complement of θ (i.e. 90O - θ) and so the force would be written in Cartesian notation as (F·Cosθ , F·Sinθ)

Adding by components

Instead of scale diagrams it is often more accurate to add vectors by using the method of components. For example, adding two vectors (with the maths convention for angles). Let the vectors be A=(A , α) and B=(B , β)

Taking components, i.e. expressing in Cartesian form, A = (A·Cosα , A·Sinα ) and B = (B·Cosβ , B·Sinβ )


The magnitude of the sum is written as:


and the angle is


The sum is thus, (|A+B|,θ).

Vector addition example 1

Add the vectors (3,13) and (4,7) then subtract (2,8) and express the result in polar form.


Vector addition example 2

Add the vectors (2,30O) and (3,45O) then subtract (1,60O), (where the angles follow the maths convention). Express the result in Cartesian form.



The Cartesian co-ordinate system has an origin and perpendicular axes. A point is space is specified by co-ordinates along each axis.
The distance between two points depends on the actual path you take.
The displacement has the length and direction of the minimum straight-line between the two points.
Polar coordinates have a straight-line length in a specified direction.
Vectors can be added by arrowed lines in scale diagrams. The arrowed lines may be positioned sequentially or simultaneously.
Vector subtraction is done by adding the negative.
Vector components are vectors at right angles which add to give the original vector.
Vectors may be added without scale diagrams by using components.

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