2.13 : Position Vectors

A position vector is a fundamental concept in mathematics that helps determine the position of one point with respect to another point in space. It is a vector that describes the direction and distance between two points. Position vectors are highly useful in the field of math and science, as they help represent spatial relationships and make calculations easier.

For instance, we want to locate a point P(x, y, z) relative to the origin of coordinates O. In that case, we can define a position vector r, which extends from the origin O to point P. We can express this vector in Cartesian vector form as: r = xi + yj + zk, where i, j, and k are the unit vectors in the x, y, and z directions, respectively. The position vector r gives us the direction and magnitude of the vector from point O to point P.

Consider a position vector directed from point A to point B in space. This vector can be denoted by the symbol r. We can also refer to this vector with two subscripts to indicate the points from and to which it is directed. Thus, we can also designate r as r_AB. Please note that if the position vectors extend from the origin of coordinates, then they are referred to only with one subscript, as r_A and r_B. The position vector r_AB can be obtained from r_A and r_B using the expression r_AB = r_B - r_A= (x_B - x_A)i + (y_B - y_A)j + (z_B - z_A)k.

For example, to establish a position vector from point A to B, the coordinates of the tail A(1 m, m0, -3 m) are subtracted from the coordinates of the head B(-2 m, 2 m, 3 m), which yields r_AB={ -3i + 2j + 6k} m.