Vectors and scalars allow us to describe physical quantities. Scalars are used for quantities that can be described completely using only one number and one unit. Vectors, on the other hand, have both a magnitude and a direction.

Examples of scalar quantities include mass, temperature, time, and density. In the case of vector quantities, two common examples are velocity and force.

## What are vectors and scalars in physics?

Vectors and scalars are two ways in which we can express physical quantities. The use of each depends on the nature of each quantity.

There are physical quantities that can be defined entirely by a single number and a unit. However, many other quantities have a direction and cannot be expressed by a single number.

A **scalar **quantity is a physical quantity that can be expressed by a single number. On the other hand, a **vector **quantity has a magnitude (the number) and a direction.

An illustration of this is the motion of a car: we must specify not only how fast it is moving, but we must also consider the direction in which it is moving.

Since vectors and scalars have different properties, we usually use letters with arrows, like $latex \vec{A}$, to indicate that vectors have direction.

## Characteristics and differences between vectors and scalars

### Main characteristics of scalars

- Scalar quantities are represented by a single number (magnitude) and a unit.
- Ordinary arithmetic operations apply to scalar quantities. For example, 3 km + 5 km = 8 km.
- Scalars are usually represented by common letters, such as A, T, S.

### Main characteristics of vectors

- Vectors are defined by a magnitude and a direction.
- Combining vectors requires a different set of operations than ordinary arithmetic.
- Vectors are usually represented by a letter with an arrow above it. For example, $latex \vec{A}, ~vec{V}$.
- Two vectors are parallel if they have the same direction.
- Two vectors are equal only if they have the same magnitude and the same direction.
- The magnitude of a vector is a scalar quantity.

## Examples of vectors and scalars

The following are some examples of vector quantities and scalar quantities:

Scalars | Vectors |

Distance | Displacement |

Time | Velocity |

Temperature | Acceleration |

Mass | Force |

Density | Momentum |

Energy |

We can understand the difference between vectors and scalars by considering distance (a scalar quantity) and displacement (a vector quantity).

Displacement is the change in the position of an object. Displacement is a vector quantity because we must specify the direction when we indicate how far an object has moved.

For example, walking 500 m north will not take us to the same place as walking 500 m south. These two displacements have the same magnitude, but different directions.

Note that **displacement** is not directly related to the total **distance** traveled.

We can represent the displacement by an arrow pointing in the direction of displacement:

The displacement is always a straight line from the starting point to the end point, i.e. it does not depend on the path taken, even if it is curved:

If an object goes to point P2 and returns to point P1, the displacement is 0. That is, the total displacement for a round trip is 0, regardless of the distance traveled:

## Vectors and scalars – Practice problems

#### Find the result of the addition of the scalar quantities: 13 kg + 15 kg.

Write the answer in the input box.

## See also

Interested in learning more about vectors? Check out these pages:

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