The **scalar magnitude** is the quantity that we can measure of a certain property that does not depend on its direction or position in space. The **vector magnitude** is the quantity that we can measure that depends on the direction or position in space.

For example, a vehicle traveling from Mexico City to Mérida will travel a **distance** of 1,323 km (scalar magnitude) but will have a **displacement** of 1007 km (vector magnitude).

Cartoon vector | Scalars | |
---|---|---|

Definition | Quantity that has magnitude and direction. | Quantity that only has magnitude. |

Sum | Vector sum or geometric resultant. | Arithmetic sum. |

Examples | Displacement, speed, weight, force. | Length, speed, mass, density, temperature. |

Index

## What is a scalar quantity?

Some physical quantities can be expressed by a number and its corresponding unit. A scalar is a quantity that has a magnitude but no direction. Scalar magnitude refers to the measure as such.

### Sum of scalar quantities

When we have two scalars of the same property, they add arithmetically as we would normally. For example: 2 kg of apples plus 4 kg of apples equals 6 kg in total.

### Examples of scalars

#### Time

Mass is the property that measures the amount of matter in a body, and also characterizes its inertial properties. The greater the mass, the greater the force must be applied to cause it to accelerate. It is a scalar quantity since the mass of a body does not change with the change of direction that that body may experience.

Mass is expressed in grams and its multiples and submultiples: milligrams, kilograms, tons, etc.

#### Distance

The distance or length is the measurement that exists between two points. It is expressed in meters or multiples thereof: millimeters, centimeters, kilometers, etc. It is a scalar quantity because the distance will be the same if we measure it from right to left or vice versa.

#### Speed

Speed is the distance an object travels in a given time. It is a scalar quantity, which when combined with direction constitutes the vector quantity that we know as velocity. For example, a swimmer who does two pools back and forth (50 mx 2 = 100 m) in 50 seconds, will have a speed of 100 m / 50 s = 2 m / s but his speed is equal to zero, since he returned to the point of departure.

## What is a vector magnitude?

Vector is the word that defines a quantity that has both direction and magnitude. The vector magnitude would be how much the vector measures.

### Vector and Vector Magnitude Representation

A vector is represented by a single bold italic letter with an up arrow:

When we want to denote the vector magnitude, we do it with the same letter that we use for the vector but in normal italics without the up arrow, or enclosed between two vertical bars:

### Sum of vector quantities

The addition of vectors requires a geometric treatment, where the vectors are represented graphically and the sum will be equal to a new or resultant vector, which will start from the initial point to the final point.

In this sense, the sum of the vector magnitudes is not equal to the arithmetic sum of the scalar magnitudes, and will depend on the magnitudes of the vectors involved and the angle they form with each other. For example, if a vehicle travels 3 km east and then heads north 3 km more, the sum of the vector magnitudes will not be 6 km but 4.24 km.

### Vector Examples

#### Displacement

An offset is simply a change in the position of a point. It is the straight segment directed from the start point to the end point, even though the path is curved. It is a vector quantity because we must express the length or distance between the two points and the direction of the change.

For example, it is not the same to say that we walk 3 km to the east than 3 km to the west; in both cases we have the same magnitude (3 km) but the directions are opposite.

#### Weight

The weight of a body is a vector quantity, it is the force exerted on the body by the attraction of the Earth. The magnitude of the weight of a body ( *w* ) of mass *m* is equal to:

where *g* is the magnitude of the acceleration due to gravity, equal to 9.8 m / s ^{2} . Weight is expressed in units of force, Newton.

#### Speed

Velocity is a physical quantity that indicates the distance a particle travels in a given time and direction. For example, a swimmer who does two pools roundtrip (50 mx 2 = 100 m) in 50 seconds, will have a speed equal to zero, since he returned to the starting point.

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