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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.4.) 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. It is conventional to always use a right-handed coordinate system. 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 (A.11)

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 (A.12)

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

According to the three-dimensional generalization of the Pythagorean theorem, the distance is given by (A.13)

By analogy, the magnitude of a general vector takes the form (A.14)

If and then it is easily demonstrated that (A.15)

Furthermore, if is a scalar then it is apparent that (A.16)   Next: Coordinate Transformations Up: Vectors and Vector Fields Previous: Vector Algebra
Richard Fitzpatrick 2016-03-31