Understanding Vector Quantities and Operations

Vector Quantities

Some physical quantities, such as force and velocity, have both direction and magnitude. These are called vector quantities. The direction must be a part of the calculations related to these quantities. Examples include a displacement of 45 meters towards the north or a velocity of 95 km/h at 30° northwest.

Characteristics of Vectors

A vector is represented graphically by an arrow, consisting of the following elements:

• Point of application: The origin of the vector.
• Magnitude (Modulus): The value of the vector, represented by the length of the arrow drawn to scale.
• Line of action: The determined path of the vector, usually given in degrees relative to a reference.
• Direction: The orientation toward which the head of the arrow points.

Types of Vectors

• Collinear Vectors: Vectors contained within the same line of action.
• Concurrent Vectors: Vectors whose lines of action intersect at a single point.
• Coplanar Vectors: Vectors contained within the same plane.
• Equal Vectors: Vectors that have the same magnitude and direction.
• Parallel Vectors: A set of vectors that have the same direction. Their lines of action are parallel, but their magnitudes may differ.
• Opposite Vector (-A): A vector is called opposite (-A) to vector A when they have the same magnitude but opposite directions.

Addition of Vectors

Adding two or more vectors results in a single resultant vector, which produces the same effect as the combined vectors. Note that vector addition is not the same as arithmetic addition. There are two primary ways to add vectors:

Graphic Methods

There are three common graphical methods to find the geometric sum of vectors:

• Triangle Method: Useful for the sum of two concurrent and coplanar vectors. Join the two vectors head-to-tail; the resultant vector forms the third side of the triangle, starting from the origin of the first vector.
• Parallelogram Method: Valid for two concurrent and coplanar vectors. Join the origins of both vectors, form a parallelogram, and the resultant vector is the diagonal originating from the common point.
• Polygon Method: Used for two or more concurrent and coplanar vectors. Join the vectors head-to-tail to form a polygon. The resultant vector connects the origin of the first to the tip of the last. If the origin of the first coincides with the end of the last, the resultant is zero, forming a "closed polygon."