082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

4.3 Cartesian coordinate system -- three dimensions 157
EXERCISES4.2
1 Plotthe following points:A(2,−2),B(−2,1),
C(−1,0),D(0, −2).
2 Statethe coordinatesofthe pointsU, V andWas
shown in Figure4.5.
3 A point Plies onthexaxis. State theycoordinate
ofP.
4 A point Qlies on theyaxis. Statethexcoordinate
ofQ.
y
y
2
2
1
V
B
1
C
–2 –1 O 1 2 3 4
–1
W
U –2
x
–2 –1 O
–1
–2
D
1
2
A
x
Figure4.5
FigureS.6
Solutions
1 FigureS.6 shows the pointsA, B,CandD.
2 U(−2, −2),V(4,1),W(3, −1)
3 0
4 0
4.3 CARTESIANCOORDINATESYSTEM--THREEDIMENSIONS
Manyengineeringproblemsrequiretheuseofthreedimensions.Figure4.6illustratesa
three-dimensional coordinate system. It comprises three axes,x,yandz. The axes are
all atright anglestoone another and intersectatthe origin, O.
The position of any point in three-dimensional space is given by specifying itsx,y
andzcoordinates.Byconventionthexcoordinateisstatedfirst,thentheycoordinateand
finally thezcoordinate. For example, P(2, 3, 4) has anxcoordinate of 2, aycoordinate
of 3 and azcoordinate of 4. It is illustrated in Figure 4.6. From the origin, P is located
by travelling 2 units in thexdirection, followed by 3 units in theydirection, followed
by 4 unitsinthezdirection.
Note that, aswith a two-dimensionalsystem,coordinates can benegative.
Example4.3 Plot the following points:A(1,−1,2), B(0,1,2), C(0,0,1).
Solution Figure 4.7 illustratesthe points A,Band C.
Wenowconsidertheequationofaplane.Anypointonthex--yplanehasazcoordinate
of 0. Hence the equation of thex--y plane isz = 0. Similarlyz = 1 represents a plane
parallel to the x--y plane but 1 unit above it. Point C in Figure 4.7 lies in the plane
z = 1.Allpoints inthis plane have azcoordinateof1.