Calculus: Analytic Geometry and Calculus, with Vectors

2.2 Coordinate systems and vectors in E3 59

n exceeds 3, the simplex does not have a plebeian name; it is an n-dimensional
simplex.

2.2 Coordinate systems and vectors in E3 To locate a point ina
plane (Euclid space E2 of two dimensions), it suffices to have a two-
dimensional rectangular coordinate system involving the two mutually
perpendicular x and y axes with which we are familiar. To locate a point

in E3 (Euclid space of three dimensions), it suffices to have a three-
dimensional rectangular coordinate system involving the three mutually
perpendicular x, y, and z axes of Figure 2.21. To partially overcome the

difficulties involved in picturing three-dimensional objects on a flat piece
of paper, we consider they and z axes to be in the plane of the paper which,
like a blackboard in a classroom, is vertical and consider the x axis to be
perpendicular to the y and z axes and sticking out toward us. We can
also consider the x and y axes to be wires on horizontal fences separating
rectangular fields and consider the z axis to be a vertical post at their
intersection.

Figure 2.21 Figure 2.22

To locate the point P(x,y,z) having nonnegative coordinates x, y, and z,
we start at the origin, go x units forward (in the direction of the positive x
axis), then go y units to the right (in the direction of the positive y axis) in
the xy plane, and then go z units upward (in the direction of the positive z
axis) to reach P(x,y,z). If x < 0, we start by going JxJ units in the direc-
tion of the negative x axis. Similar rules apply when other coordinates
are negative. Figure 2.21 shows the point P(3,3,3) and, in addition, the
projections Px, P, PZ, Qz, Q, and QZ of this point on the three coordinate
planes and coordinate axes. The figure is worth a little study. The
eight encircled points lie at the vertices of a cube. Each of the edges is 3
units long, but in the flat figure the distance between two points 1 unit
apart on the x axis is only a half or a third or a quarter of the distance
between two such points on the y and z axes. Further information about
the natures of figures involving rectangular coordinate systems in E3 can
be obtained by looking at Figure 2.22. This shows a sphere with center