## Introduction to Vectors and Scalars

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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**.

## 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: 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 \(\text{500}\) \(\text{g}\). The mass of the tub of margarine is a scalar quantity. It only needs one number to describe it, in this case, \(\text{500}\) \(\text{g}\).

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 *and* **which direction** it is in.

### Definition: 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 \(\text{100}\) \(\text{km·h$^{-1}$}\). What we have here is a vector called the velocity. The car is moving at \(\text{100}\) \(\text{km·h$^{-1}$}\) (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.

## 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 tutorial, vectors will be shown by symbols with an arrow pointing to the right above it. For example, \(\vec{F}\), \(\vec{W}\) and \(\vec{v}\) 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:

- \(\vec{F}\) represents the force vector
- \(F\) represents the magnitude of the force vector

## Optional Video: Introduction to Vector and Scalars

This video by Khan Academy below introduces vectors and scalars.