Appendix B page 1B.1 VECTORS AND SETS Key ideas: orthogonal vector, hyper-planes, convex sets, open and closed sets We now extend our analysis to ordered n-tuples or “vectors.” Each component of the vector 1(,...,)nxxx=is a real number. Where it is important to make clear the dimension of the vector, we write nx∈\. If each component is positive, then xis positive and we write nx+∈\. We will describe the vector yas being larger than the vector xif every component of yis at least as large as xand write yx≥. If at least one component of yis strictly larger we will say that yis strictly larger than xand write yx>. (We will occasionally use the notation xy±to indicate that all components are strictly larger.) A neighborhood of a vector is the set of points near it. Thus to define a neighborhood we need some measure of distance ( , )d y zbetween any two vectors 1(,...,)nyyy=and 1(,...,)nzzz=. If 1n=, a natural measure is the absolute value of the difference so that ( , )d y zyz=−. Similarly, with 2n=, a natural measure is the Euclidean distance yz−between the two vectors. Appealing to Pythagoras Theorem, 2221122()()yzyzyz−=−+−. Extending this to n-dimensions, the square of Euclidean distance between ordered n-tuples is defined as follows. 22yz−11yz−yz−1x2xFig. B.1-1: Distance between 2 vectors yz
Appendix B page 2221()njjjyzyz=−=−∑It is sometimes more convenient to express distance as a vector product. Vector Product The product (“inner product” or “sumproduct”) of two n-dimensional vectors yand zis. 1niiiy zy z=⋅=∑From this definition, it follows that the distributive law for multiplying vectors is the same as for numbers. Distributive law for vector multiplication ()abca ba c⋅+=⋅+⋅Orthogonal Vectors Two vectors, xand pare depicted below in 3-dimensional space. Fig. B.1-2: Orthogonal Vectors 1xO x2x3xxpppx−
has intentionally blurred sections.
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