Scalar And Vector Quantities - Types of Vectors
Some quantities can be represented fully just by their magnitude or numerical size. However, others require additional information about the direction to be described appropriately. This distinction becomes very important when formulating equations and doing analysis.
Scalars Quantities
Scalars are quantities that only have magnitude. They can be completely specified by stating a numerical value and the units. Mass, time, temperature, volume, density, and energy are all examples of scalar quantities commonly used in engineering. For instance, saying a pipe length of 2 meters fully describes scalar quantity. Scalars follow the standard rules of algebra and arithmetic when mathematical operations are applied. They can be added, subtracted, multiplied or divided like regular numbers. Graphically representing scalars, they are shown as number lines or coordinate points without arrows.
Vector Quantities
Vectors are quantities that possess both magnitude and direction. To fully describe a vector, you need to state its numerical magnitude and provide a direction. Some examples of vectors used in engineering include force, velocity, acceleration, momentum, and displacement. For instance, a 60 km/hr velocity to the north would require stating both the magnitude (60 km/hr) and the direction (north) to properly define it as a vector.
Vectors are represented mathematically by boldfaced letters like A. In handwriting it is written as A with an arrow on top, A ⃗. The magnitude of a vector uses the same letter but in the form of |A| or |A ⃗ |. Graphically, vectors are depicted as arrows, with the arrowhead indicating direction and the length displaying magnitude to scale. Vectors follow specific vector addition and subtraction rules, which involve geometrical approaches.
Types of Vectors
Vectors can be categorized based on certain properties. Some important vector types include:
Zero Vector - A vector with zero magnitude. Represented by a point or an arrow with no length.
Unit Vector - A vector with a magnitude of exactly one. Denoted with a hat (^) symbol. Useful for representing direction.
Position Vector - A vector drawn from the origin of a coordinate system to a point of interest. Defines the position of the point.
Co-Initial Vectors - Vectors that start from the same initial point.
Like and Unlike Vectors - Vectors with the same or opposite directions are like and unlike vectors.
Co-Planar Vectors - Vectors that lie in the same plane.
Collinear Vectors - Vectors aligned along the same straight line.
Equal Vectors - Vectors having precisely the same magnitude and direction.
Displacement Vector - A vector representing the change in position of an object.
Negative of a Vector - Flips the direction of a vector while keeping the same magnitude. Typically, it indicates an opposite sense.
Orthogonal Vectors - Two vectors are considered orthogonal or perpendicular if they are at a 90 degree angle with respect to each other. This means the direction of one vector is perpendicular to the direction of the other vector. In visual terms, if one vector was horizontal and the other was vertical, they would depict orthogonal vectors.
Concurrent Vectors - Concurrent vectors refer to multiple vectors that share a common point of intersection. In other words, a set of concurrent vectors all pass through the same point in space. This common point is where the vector lines of action meet.
These vector classifications allow us to characterize key properties. Understanding these categories is crucial for properly analyzing vectors in areas like physics, engineering, and computer graphics. Visualizing and manipulating these different vector types aids in solving spatial problems.