Engineering Calculus Notes 118

Engineering Calculus Notes 118 - 106 CHAPTER 1 COORDINATES...

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Unformatted text preview: 106 CHAPTER 1. COORDINATES AND VECTORS z Q′ O P′ R′ → − v O x Q R y P Figure 1.54: OP RQ −→ −→ → −− We see from Figure 1.54 that the vectors OP , OQ, − obey the right-hand v rule, so have positive orientation. Given any four points A, B , C , and D in R3 , we can form a “pyramid” built on the triangle △ABC , with a “peak” at D (Figure 1.55). The D C A B Figure 1.55: Simplex △ABCD traditional name for such a solid is tetrahedron, but we will follow the terminology of combinatorial topology, calling this the simplex22 with vertices A, B , C and D , and denote it △ABCD ; it is oriented when we pay attention to the order of the vertices. Just as for a triangle, the edges emanating from the vertex A are represented by the displacement vectors −− − →→ −→ − AB , AC , and AD . The first two vectors determine the oriented “base” triangle △ABC , and the simplex △ABCD is positively (resp. negatively) oriented if the orientation of △ABC is positive (resp. negative) when 22 Actually, this is a 3-simplex. In this terminology, a triangle is a 2-simplex (it lies in a plane), and a line segment is a 1-simplex (it lies on a line). ...
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