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Cartesian Components of a Vector

Consider a Cartesian coordinate system consisting of an origin, , and three mutually perpendicular coordinate axes, , , and --see Figure A.99. Such a system is said to be right-handed if, when looking along the direction, a clockwise rotation about is required to take into . Otherwise, it is said to be left-handed. In physics, it is conventional to always use right-handed coordinate systems. It is convenient to define unit vectors, , , and , parallel to , , and , respectively. Incidentally, a unit vector is a vector whose magnitude is unity. The position vector, , of some general point whose Cartesian coordinates are ( , , ) is then given by (1271)

In other words, we can get from to by moving a distance parallel to , then a distance parallel to , and then a distance parallel to . Similarly, if is an arbitrary vector then (1272)

where , , and are termed the Cartesian components of . It is coventional to write . It follows that , , and . Of course, .

According to the three-dimensional generalization of the Pythagorean theorem, the distance is given by (1273)

By analogy, the magnitude of a general vector takes the form (1274)

If and then it is easily demonstrated that (1275)

Furthermore, if is a scalar then it is apparent that (1276)   Next: Coordinate Transformations Up: Vector Algebra and Vector Previous: Vector Algebra
Richard Fitzpatrick 2011-03-31