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Unformatted text preview: Chapter 2 3Space: lines and planes In this chapter we discuss how to describe points, lines and planes in 3space. introduce the language of vectors. discuss various matters concerning the relative position of lines and planes: in par- ticular, intersections and angles. show how to translate and rotate lines and planes in relatively simple cases. 2.1 3space and vectors 2.1.1 Coordinatizing 3space The physical world around us is called threedimensional because through any point pre- cisely three mutually perpendicular axes (and no more) can pass. Such axes can be used to describe points in 3-space by triples of numbers: the (signed) distances to planes formed by two of these axes. We call such a set of perpendicular axes through a given point a cartesian coordinate system . Each of the axes is called a coordinate axis and the point of intersection of the three axes is called the origin . Every point of a coordinate axis cor- responds to a real number. As you know, often, but not necessarily always, these axes are called x axis, y axis and z axis. By convention, we agree that the coordinate sys- tem be righthanded: if you turn the positive x axis to the positive y axis, then a screw positioned along the z axis following this movement would move in the direction of the positive z axis. Every two of the three axes span a plane: the y,z plane, the x,z plane and the x,y plane, respectively. The signed distances u , v and w of a point P in 3space to each of these planes in the given order are the coordinates of P ; we usually put the three together as follows: ( u,v,w ) and still call this the coordinates of P . Notations for points that occur often: ( x,y,z ), ( x 1 ,x 2 ,x 3 ), ( a,b,c ), ( a 1 ,a 2 ,a 3 ), ( u,v,w ), etc. For a point P with coordinates ( x,y,z ) we also write P = ( x,y,z ). 21 22 3Space: lines and planes x z y 4 P=(2,3,4) x y z Figure 2.1: Left: The third coordinate of P = (2 , 3 , 4) is the signed distance of P to the x,y plane; so if the point were below the x,y plane, the third coordinate would have been negative. Right: Cartesian coordinate systems are taken to be righthanded: a screw positioned along the z axis moves along the positive z axis if turned in the direction from x axis to y axis. Most discussions in this chapter will focus on 3space. But with some adaptations (usually simplifications) the material makes sense in 2space. 2.1.2 Example. Given the point P with coordinates ( x,y,z ), the point Q with coordinates ( x,y, 0) is the vertical projection of P on the x,y plane. The distance between P and Q is | z | (note the absolute value). Similarly, the (horizontal) projection on the y,z plane has coordinates (0 ,y,z ). Finally, the horizontal projection on the x,z plane has coordinates ( x, ,z )....
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