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 we can measure that depends on the direction or position in space.
For example, a vehicle traveling from Mexico City to Mérida will travel one distance of 1323 km (scalar magnitude) but it will have a displacement of 1007 km (vector magnitude).
| Vectors | Scalars | |
|---|---|---|
| Definition | Quantity having magnitude and direction. | Quantity having only magnitude. |
| Sum | Vector sum or geometric resultant. | Arithmetic sum. |
| Examples | Displacement, velocity, weight, force. | Length, speed, mass, density, temperature. |
What is a scalar magnitude
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 measurement as such.
Sum of scalar magnitudes
When we have two scalars of the same property, they are added arithmetically as we would normally do. For example: 2 kg of apples plus 4 kg of apples equals 6 kg in total.
Examples of scalars
Mass
Mass is the property that measures the amount of matter in a body, and also characterizes the inertial properties of the body. The greater the mass, the greater the force that must be applied to cause acceleration of the body. It is a scalar quantity because the mass of a body does not change with the change of direction that the body may experience.
Mass is expressed in grams and its multiples and submultiples: milligrams, kilograms, tons, etc.
Distance
Distance or length is the measure that exists between two points. It is expressed in meters or multiples of it: millimeters, centimeters, kilometers, etc. It is a scalar magnitude in that the distance will be equal 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 known as velocity. For example, a swimmer who swims two pools round trip (50 m x 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, because he returned to the starting point.
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.
Representation of vector and vector magnitude.
A vector is represented by a single letter in bold italics with an arrow above:
When we want to denote the vector magnitude we do it with the same letter we use for the vector but in normal italics without the arrow above, or enclosed between two vertical bars:
Sum of vector magnitudes
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 between them. For example, if a vehicle travels 3 km to the east and then heads north 3 km more, the sum of the vector magnitudes will not be 6 km but 4.24 km.
Examples of vectors
Displacement
A displacement is simply a change in the position of a point. It is the straight segment directed from the initial point to the end point, even if the trajectory is curved. It is a vector quantity in that we must express the length or distance between the two points and the direction of 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 acceleration due to gravity, equal to 9.8 m/s2. Weight is expressed in units of force, Newton.
Velocity
Velocity is a physical quantity that indicates the displacement distance of a particle in a given time and direction. For example, a swimmer who swims two pools round trip (50 m x 2= 100 m) in 50 seconds, will have a velocity equal to zero, because he returned to the starting point.
You may be interested to know the differences between:
