FHSST Physics/Vectors/Introduction

Introduction

A vector is 'something' that has both magnitude and direction. "'Thing'? What sorts of 'thing'?" Any piece of information which contains a magnitude and a related direction can be a vector. A vector should tell you how much and which way.

Consider a man driving his car east along a highway at 100 km/h. What we have given here is a vector — the car's velocity. The car is moving at 100 km/h (this is the magnitude) and we know where it is going — east (this is the direction). Thus, we know the speed and direction of the car. These two quantities, a magnitude and a direction, form a vector we call velocity.

Definition: A vector is a measurement which has both magnitude and direction.

In physics, magnitudes often have directions associated with them. If you push something it is not very useful knowing just how hard you pushed. A direction is needed too. Directions are extremely important, especially when dealing with situations more complicated than simple pushes and pulls.

Different people like to write vectors in different ways. Any way of writing a vector so that it has both magnitude and direction is valid.

Are vectors physics? No, vectors themselves are not physics. Physics is just a description of the world around us. To describe something we need to use a language. The most common language used to describe physics is mathematics. Vectors form a very important part of the mathematical description of physics, so much so that it is absolutely essential to master the use of vectors.

Mathematical representation

Numerous notations are commonly used to denote vectors. In this text, vectors will be denoted by symbols capped with an arrow. As an example, ${\overrightarrow {s}}$, ${\overrightarrow {v}}$ and ${\overrightarrow {F}}$are all vectors (they have both magnitude and direction). Sometimes just the magnitude of a vector is required. In this case, the arrow is omitted. In other words, F denotes the magnitude of vector ${\overrightarrow {F}}$. $|{\overrightarrow {F}}|$ is another way of representing the size of a vector.

Graphical representation

Graphically vectors are drawn as arrows. An arrow has both a magnitude (how long it is) and a direction (the direction in which it points). For this reason, arrows are vectors.

In order to draw a vector accurately we must specify a scale and include a reference direction in the diagram. A scale allows us to translate the length of the arrow into the vector's magnitude. For instance if one chose a scale of 1cm = 2N (1cm represents 2N), a force of magnitude 20N would be represented as an arrow 10cm long. A reference direction may be a line representing a horizontal surface or the points of a compass.

Worked Example 2: Drawing vectors

Question: Using a scale of $1cm=2m.s^{-1}$ represent the following velocities:

a) $6m.s^{-1}$ north

b) $16m.s^{-1}$ east

Answer: