Differences between Scalars and Vectors
In a nutshell, scalar quantities have only magnitude and no direction, while vector quantities have both magnitude and direction.
Here are the details of their differences:
| Characteristic | Scalar Quantity | Vector Quantity |
|---|---|---|
| Definition | A scalar quantity has only magnitude (size) and no direction. | A vector quantity has both magnitude and direction. |
| Example | Mass, time, temperature, speed, distance, energy, pressure | Displacement, velocity, acceleration, force, moment |
| Representation | Typically represented by a single number or value with units, e.g. m = 5 kg (mass) t = 10 s (time) | Represented by both magnitude and direction, often using bold letters or arrows, e.g. $\vec{v}$ =20 m/s north (velocity) $\textbf{F}$ = 50 N east (force) |
| Arithmetic Operations | Scalars can be added, subtracted, multiplied, or divided by scalars to yield scalars (e.g., adding temperatures or multiplying speed by time). | Vectors can be added or subtracted only by vectors of the same kind, and the result is also a vector. Scalar multiplication of a vector results in a vector (e.g., adding velocities or multiplying force by time). |
Addition of Vectors
When adding vectors, you place them head-to-tail in a geometric fashion, creating a resultant vector that points from the tail of the first vector to the head of the last vector.
If the vectors are in the same direction, the resultant vector is also in the same direction and its magnitude is merely the addition of the magnitudes of the vectors being added.
If the vectors are in different directions, the magnitude and direction of the resultant vector can be determined using vector addition rules, such as the parallelogram rule or the triangle rule. Once the vector diagrams are drawn, the magnitude of the resultant vector can be determined by scale drawing. The direction of the resultant vector can also be measured relative to another vector using a protractor.
The following video shows an example of how scale drawing is used to find the resultant of two vectors.
The following applet allows you to change the direction of two vectors. Drag one of the vectors and place it at the end of the other to observe how the parallelogram rule or triangle rule works.
Subtraction of Vectors
Subtraction of vectors is essentially the addition of the negative of a vector, where you reverse the direction of the vector you want to subtract and then follow the same vector addition process.
