Understanding Engineering Mathematics

Answers

1. (i)e=−(a+b+c+d) (ii)

−→

CE=−(e+a+b) (iii)

−→

CE=c+d

(iv)

−→

EC=

−→

EA−

−→

CA

2.b−ais of length

√

3 parallel to they-axis

11.4 Rectangular Cartesian coordinates in three dimensions

To represent points, geometrical objects, vectors, etc. in 3-dimensional space it is useful to
have areference frameorcoordinate system. The simplest type is the Cartesian system
of rectangular coordinates, which generalises the usualx-,y-axes of two dimensions – see
Figure 11.6, which shows how a pointP may be represented by coordinates(x 1 ,y 1 ,z 1 )
relative to a three-dimensional rectangular system ofx-,y-,z-axes.

P(x 1 , y 1 , z 1 )

x

z 1

y 1

y

N

Qx 1

z

0

r =

OP

=^

√x^1

2 +x

22 +

x^3

2

Figure 11.6Rectangular Cartesian coordinates in three dimensions.

PNis the perpendicular to thex-yplane. The axes are shown arranged in the usual
convention in which theOx,Oy,Ozform aright-handed set– rotatingOxround to
Oydrives a right-handed screw alongOz. The same applies if we replaceOx→Oy→
Oz→Ox,etc.
We say that the coordinates ofPrelative to the axesOxyzare(x 1 ,y 1 ,z 1 )or ‘Pis the
point(x 1 ,y 1 ,z 1 )’ where the coordinates are the distances ofP along thex-,y-,z-axes
respectively. We can label each point of 3-dimensional space with a set of ‘coordinates’
(x, y, z)relative to a particular coordinate system. For a given point, choosing a different
set of axes gives a different set of coordinates.

Exercise on 11.4

Plot the points (0, 0, 0), (0, 0, 1), (0, 1, 0), (1, 0, 0) (−1, 0, 0), (1, 0,−1), (2,−1, 1) on
a perspective drawing of a Cartesian rectangular coordinate system.