Definition
A vector can be represented in different ways without changing its physical meaning. Regardless of the representation used, the vector always has the same magnitude and direction.
Different representations are useful for different purposes. A graphical representation helps visualize a vector, while numerical representations are more convenient for calculations.
Arrow Representation
The most common way to represent a vector is by drawing an arrow.
- The length of the arrow represents the magnitude.
- The arrowhead indicates the direction.
For example, two arrows with the same length and pointing in the same direction represent the same vector, even if they are drawn at different locations.
Magnitude and Direction Representation
A vector can also be described by stating its magnitude and direction.
Examples:
- 5 m east
- 20 N upward
- 12 m/s at 35° above the horizontal
This representation provides all the information needed to define the vector without drawing it.
Coordinate Representation
A vector may also be represented by its coordinates.
Examples:
This form specifies the vector’s coordinates along the x- and y-axes. The relationship between coordinates and vector components is explained in the next section.
Key Points
- A vector can be represented in different forms.
- All valid representations describe the same vector.
- The choice of representation depends on the purpose of the problem.
- Coordinate representation is especially useful for calculations and will be discussed in detail in the next section.