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Unformatted text preview: 106 CHAPTER 1. COORDINATES AND VECTORS
Q′ O P′ R′ →
v O x Q
y P Figure 1.54: OP RQ −→ −→ →
We see from Figure 1.54 that the vectors OP , OQ, − obey the right-hand
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
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
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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