Position Vector

Position Vector

  • A position vector denotes the position of a point in space with respect to some reference origin.
  • It is typically represented by a ray pointing from the origin to the point of interest in a coordinate plane or 3-dimensional coordinate system.
  • It’s called a ‘vector’ because it has both magnitude and direction; its ‘position’ is determined by the direction and length of the arrow.
  • A position vector is generally denoted as r = (a, b) in 2D or r = (a, b, c) in 3D, where (a, b, c) are the coordinates of the point.
  • Note that while a position vector refers to the position of a discrete point in space, it is not tied to one specific point in space.

Characteristics of Position Vectors

  • The origin has a position vector of zero, denoted as r = (0,0) or r = (0,0,0). This is because from the origin to the origin, there is no displacement, so both magnitude and direction are undefined or zero.
  • A position vector is said to be in the same direction as another if one is a scalar multiplication of the other.
  • If r = (a, b, c) is a position vector, then the position vector in the opposite direction is -r = (-a, -b, -c).
  • Two position vectors r1 and r2 are equal if their corresponding coordinates are equal, i.e., if r1 = (a1, b1, c1) and r2 = (a2, b2, c2), then a1 = a2, b1 = b2, and c1 = c2.

Relative Position Vectors

  • The relative position vector of one point with respect to another is the vector displacement from the first point to the second.
  • In other words, it is the vector that points from the first point to the second point.
  • The relative position vector from point A (with position vector a) to B (with position vector b) is given by b - a.

Applications of Position Vectors

  • Understanding position vectors is integral to many branches of physics and engineering, including mechanics, electromagnetism, and celestial mechanics.
  • They provide the basic language for describing the positions of particles, objects, or systems of objects in space, facilitating the study of motion, forces, energy, and other concepts.
  • Position vectors also play a crucial role in geometry, helping to define distances, angles, planes, and other geometric properties.