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Introduction to vectors and scalars

We come into contact with many physical quantities in the natural world on a daily basis. For example, things like time, mass, weight, force, and electric charge, are physical quantities with which we are all familiar. We know that time passes and physical objects have mass. Things have weight due to gravity. We exert forces when we open doors, walk along the street and kick balls. We experience electric charge directly through static shocks in winter and through using anything which runs on electricity.

There are many physical quantities in nature, and we can divide them up into two broad groups called vectors and scalars.

Chapter introduction

Scalars and vectors

Scalars are physical quantities which have only a number value or a size (magnitude). A scalar tells you how much of something there is.

Definition 1: Scalar

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

For example, a person buys a tub of margarine which is labelled with a mass of 500 g. The mass of the tub of margarine is a scalar quantity. It only needs one number to describe it, in this case, 500 g.

Vectors are different because they are physical quantities which have a size and a direction. A vector tells you how much of something there is andwhich direction it is in.

Definition 2: Vector

A vector is a physical quantity that has both a magnitude and a direction.

For example, a car is travelling east along a freeway at 100 km·h−1. What we have here is a vector called the velocity. The car is moving at 100 km·h−1 (this is the magnitude) and we know where it is going – east (this is the direction). These two quantities, the speed and direction of the car, (a magnitude and a direction) together form a vector we call velocity.

Examples of scalar quantities:

  • mass has only a value, no direction

  • electric charge has only a value, no direction

Examples of vector quantities:

  • force has a value and a direction. You push or pull something with some strength (magnitude) in a particular direction

  • weight has a value and a direction. Your weight is proportional to your mass (magnitude) and is always in the direction towards the centre of the earth.

Exercise 1: Vectors and scalars

Classify the following as vectors or scalars

  1. length

  2. force

  3. direction

  4. height

  5. time

  6. speed

  7. temperature

  1. scalar
  2. vector
  3. scalar
  4. scalar
  5. scalar
  6. vector
  7. scalar

Vector notation

Vectors are different to scalars and must have their own notation. There are many ways of writing the symbol for a vector. In this book vectors will be shown by symbols with an arrow pointing to the right above it. For example, F , W and v represent the vectors of force, weight and velocity, meaning they have both a magnitude and a direction.

Sometimes just the magnitude of a vector is needed. In this case, the arrow is omitted. For the case of the force vector:

  • F represents the force vector

  • F represents the magnitude of the force vector