Position Vectors

A position vector is a fixed vector that locates the position of a point in space relative to another point. It describes the direction and distance between two points. For example, a position vector r directed from the origin O to point P(x, y, z) can be expressed in Cartesian vector form as r = xi + yj + zk, where i, j, and k are unit vectors along the x, y, and z axes.

A position vector in Cartesian vector form uses unit vectors i, j, and k to represent components along the x, y, and z axes. For a point P with coordinates (x, y, z), the position vector is r = xi + yj + zk. This notation clearly shows the magnitude of the vector in each coordinate direction and allows for precise mathematical calculations.

Position vectors from the origin are designated with one subscript, such as rA or rB. Position vectors between two points use two subscripts, like rAB, indicating direction from point A to point B. The vector rAB is calculated by subtracting the coordinates of the tail point A from the head point B.

To calculate a position vector rAB from point A to point B, subtract the coordinates of A from the coordinates of B using the formula: rAB = rB - rA = (xB - xA)i + (yB - yA)j + (zB - zA)k. For example, with A(1, 0, -3) and B(-2, 2, 3), the result is rAB = -3i + 2j + 6k meters.

Unit vectors i, j, and k represent the directional components along the x, y, and z axes respectively. They establish the Cartesian coordinate system framework and allow position vectors to be decomposed into their directional components. Each unit vector has a magnitude of one and points along its respective axis.

The triangle rule states that a resultant position vector from point A to point B can be obtained by expressing the individual position vectors rA and rB in Cartesian form, then subtracting rA from rB. This geometric method visualizes how two vectors from the origin combine to create the vector between two arbitrary points in space.

Position vectors help represent spatial relationships and make calculations easier in mathematics and science. They enable engineers to locate points in three-dimensional space relative to reference frames, which is essential for analyzing forces, motion, and structural geometry in mechanical systems and design problems.