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Handbook of mathematics for engineers and scienteists part 22

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The straight lines connecting the vertices of a triangle with the points at which the respective opposite sides are tangent to the excircles intersect in a single point called the Nagelpoint (Fig. 3.5). | 4.5. Coordinates Vectors Curves and Surfaces in Space 115 Figure 4.29. Point in rectangular Cartesian coordinate system. of intersection of the perpendiculars with the axes by Mx My and Mz respectively. The numbers see Fig. 4.29 x OMx y OMy z OMz 4.5.2.1 where OMx OMy and OMz are the signed lengths of the directed segments oMx OMy and oMz of the axes OX OY and OZ respectively are called the coordinates of the point M in the rectangular Cartesian coordinate system. The number x is called the first coordinate or the abscissa of the point M the number y is called the second coordinate or the ordinate of the point M and the number z is called the third coordinate or the applicate of the point M. Usually one says that the point M has the coordinates x y z and the notation M x y z is used. To each point M of three-dimensional space one can assign its position vector. The directed segment oM is called the position vector of the point M. The position vector determines the vector r r oM whose coordinates are its projections on the axes OX OY and OZ respectively. Obviously the triple x y z of numbers can be called the point M whose coordinates are these numbers or the position vector oM whose projections are these numbers. An arbitrary vector x y z can be represented as x y z xi yj zk 4.5.2.2 where i 1 0 0 j 0 1 0 and k 0 0 1 are the unit vectors with the same directions as the coordinate axes OX OY and OZ. The distance between points M1 x1 y1 z1 and M2 x2 y2 zf is determined by the formula d x2 - X1 2 y2 - y 1 2 Z2 - Z1 2 rf - r1 4.5.2.3 where r2 0M2 and r1 M1 are the position vectors of the points M1 and M2 respectively see Fig. 4.30 . Figure 4.30. Distance between points. 116 Analytic Geometry Two vectors a x1 yi z1 and b x2 y2 z2 are equal to each other if and only if the following relations hold simultaneously xi x2 yi y2 zi z2. For arbitrary vectors one has the relations xi yi Z1 X2 y2 Z2 xi X2 yi y2 Z1 z2 a x y z ax ay az . 4.5.2.4 If a point M divides the directed .