Position Vectors

Position Vectors

Definition

• A position vector describes the position of a point in space relative to an origin.
• It is a vector that starts from a fixed point, usually the origin (0,0) in 2D or (0,0,0) in 3D, and ends at the point under consideration.

Notation and Description

• The position vector of a point A is often denoted by OA.
• It is represented by the coordinates of the point A with respect to the origin.
• In two-dimensions, a position vector would be ai + bj; whereas, in three dimension, it would be ai + bj + ck.

Operations

• Position vectors can be added, subtracted, and multiplied by scalars, just like any other vectors.
• When subtracting two position vectors, the order matters: the vector BA points from A to B, while AB points from B to A.

Distance between Two Points

• The distance between two points A and B, can be found using their position vectors. The vector AB can be found by subtracting OA and OB (i.e., AB = OB - OA) and the distance can be found by calculating the magnitude of AB.

Importance

• Position vectors are fundamental in vector algebra and physics as they allow for a concrete understanding of points in space.
• They pave the way for describing geometrical transformations, such as rotation, translation, and scaling.

Applications

• Analytical geometry makes extensive use of position vectors to define geometric objects such as lines and planes.
• In Physics, position vectors are crucial for understanding concepts such as displacement, velocity, and acceleration.
• In computer graphics, they play a key role in representing objects in a 3D space.