By Arthur Gill
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Additional resources for Applied algebra for the computer sciences
3 ). It is important to keep in mind that the components are to be expressed in the same set of unit vectors. This typically means setting up mutually perpendicular 13We assume that the reader is familiar with the symbol I: simply as a summation symboL We shall discuss its properties and ways of manipulating it in Chapter 5. 3 VECTO RS IN 0 I FFE RENT COO RD I NATE SYSTEMS 21 unit vectors (an orthonormal basis) at a single point and resolving all vectors along those unit vectors. The dot product, in various forms and guises, has many applications in physics.
Example. To illustrate further the conversion of vectors from one coordinate system to another, consider a charge q that is located at the cylindrical coordinates (a, 11:/3, -a). We want to find the spherical components of the electrostatic field E of this charge at a point P with Cartesian coordinates (a, a, a). 4 RELATIONS AMONG UNIT VECTORS 37 The most straightforward way of doing this is to convert all coordinates to Cartesian, find the field, and then take the dot products with appropriate unit vectors.
Ql52er . 2268er . 1724 2keq , a Eo = E . 0567eo . 0152eo . 2268eo . 1558 2 , a Erp a = E . 0567erp . 0152erp . 2268erp . 0294 2keq . a The choice of Cartesian coordinates was the most straightforward one, but one can choose any other coordinate system to calculate the field and find the components in any • other set of unit vectors. The reader is urged to try the other choices. 1. Find the equation of a line that passes through the following pairs of points: (a) (1,0,1) and (-1,1,0). (b) (2,2, -1) and (-2, -1, 1).
Applied algebra for the computer sciences by Arthur Gill