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Cartesian Components of a Vector

Consider a Cartesian coordinate system $ Oxyz$ , consisting of an origin, $ O$ , and three mutually perpendicular coordinate axes, $ Ox$ , $ Oy$ , and $ Oz$ . (See Figure A.4.) Such a system is said to be right-handed if, when looking along the $ Oz$ direction, a $ 90^\circ $ clockwise rotation about $ Oz$ is required to take $ Ox$ into $ Oy$ . Otherwise, it is said to be left-handed. It is conventional to always use a right-handed coordinate system.

Figure A.4: A right-handed Cartesian coordinate system.
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It is convenient to define unit vectors, $ {\bf e}_x$ , $ {\bf e}_y$ , and $ {\bf e}_z$ , parallel to $ Ox$ , $ Oy$ , and $ Oz$ , respectively. Incidentally, a unit vector is a vector whose magnitude is unity. The position vector, $ {\bf r}$ , of some general point $ P$ whose Cartesian coordinates are ($ x$ , $ y$ , $ z$ ) is then given by

$\displaystyle {\bf r} = x\,{\bf e}_z + y\,{\bf e}_y+z\,{\bf e}_z.$ (A.11)

In other words, we can get from $ O$ to $ P$ by moving a distance $ x$ parallel to $ Ox$ , then a distance $ y$ parallel to $ Oy$ , and then a distance $ z$ parallel to $ Oz$ . Similarly, if $ {\bf a}$ is an arbitrary vector then

$\displaystyle {\bf a} = a_x\,{\bf e}_x+ a_y\,{\bf e}_y+a_z\,{\bf e}_z,$ (A.12)

where $ a_x$ , $ a_y$ , and $ a_z$ are termed the Cartesian components of $ {\bf a}$ . It is conventional to write $ {\bf a} \equiv (a_x,\,a_y,\,a_z)$ . It follows that $ {\bf e}_x\equiv (1,\,0,\,0)$ , $ {\bf e}_y\equiv (0,\,1,\,0)$ , and $ {\bf e}_z\equiv
(0,\,0,\,1)$ . Of course, $ {\bf0} \equiv (0,\,0,\,0)$ .

According to the three-dimensional generalization of the Pythagorean theorem, the distance $ OP\equiv \vert{\bf r}\vert=r$ is given by

$\displaystyle r = \sqrt{x^{\,2} + y^{\,2} + z^{\,2}}.$ (A.13)

By analogy, the magnitude of a general vector $ {\bf a}$ takes the form

$\displaystyle a = \sqrt{a_x^{\,2} + a_y^{\,2} + a_z^{\,2}}.$ (A.14)

If $ {\bf a} \equiv (a_x,\,a_y,\,a_z)$ and $ {\bf b}\equiv (b_x,\,b_y,\,b_z)$ then it is easily demonstrated that

$\displaystyle {\bf a} + {\bf b} \equiv (a_x+b_x,\,a_y+b_y,\,a_z+b_z).$ (A.15)

Furthermore, if $ n$ is a scalar then it is apparent that

$\displaystyle n\,{\bf a} \equiv (n\,a_x,\,n\,a_y,\,n\,a_z).$ (A.16)


next up previous
Next: Coordinate Transformations Up: Vectors and Vector Fields Previous: Vector Algebra
Richard Fitzpatrick 2016-03-31