PreCal 10.1 The 3D Coordinate System ­ End.notebook
10-1
March 29, 2017
The 3-D Coordinate System
You can construct a 3­D coordinate system by passing a z­axis perpendicular to both the x­ and y­axes at the origin. Taken as pairs, the axes determine three coordinate planes: the xy­plane, the xz­plane, and the yz­plane. In the 3­D system, a point P in space is determined by and ordered triple (x, y, z) where x, y, and z are as follows.
x = directed distance from the yz­plane to P y = directed distance from the xz­plane to P
z = directed distance from the xy­plane to P
Octants
These three coordinate planes separate the 3­D coordinate system into eight octants. The first octant is the one in which all three coordinates are positive. A 3­D coordinate system can have either a left­handed or a right­
handed orientation. In this class, you will work exclusively with right­
handed systems. In a right­handed system, Octants II, III, and IV are found by rotating counterclockwise around the positive z­axis. Octant V is vertically below Octant I. Octants VI, VII, and VIII are then found by rotating counterclockwise around the negative z­axis.
1
PreCal 10.1 The 3D Coordinate System ­ End.notebook
March 29, 2017
Ex. 1 Plot each point in space.
A.
(2, −3, 3)
B.
(−2, 6, 2)
z
z
y
y
x
x
C.
(1, 4, 0)
D.
(2, 2, −3)
z
z
y
y
x
x
The Distance and Midpoint Formulas
Many of the formulas established for the two­dimensional coordinate system can be extended to three dimensions.
2
PreCal 10.1 The 3D Coordinate System ­ End.notebook
March 29, 2017
Ex. 2 Find the distances between the points.
A.
(0, 1, 3) and (1, 4, −2) B.
(−1, −4, 1) and (2, 5, −6)
Ex. 3 Find the midpoint of the line segment joining the two points.
A.
(5, −2, 3)
and
(0, 4, 4)
B.
(2, −2, 3)
and
(1, 3, 6)
3
PreCal 10.1 The 3D Coordinate System ­ End.notebook
March 29, 2017
The Equation of a Sphere
A sphere with center (h, k, j) and radius r is defined as the set of all point (x, y, z) such that the distance between (x, y, z) and (h, k, j) is r. Using the Distance Formula we get the following equation:
r
By squaring each side of this equation, you obtain the standard equation of a sphere.
Ex. 4 Find the standard equation of the sphere with center (2, 4, 3) and radius 3. Does this sphere intersect the xy­plane?
4
PreCal 10.1 The 3D Coordinate System ­ End.notebook
March 29, 2017
Ex. 5 Find the center and radius of the sphere. A.
x2 + y2 + z2 + 4x − 2y + 8z + 10 = 0 B.
x2 + y2 + z2 − 2x + 4y − 6z + 8 = 0 Trace of a Surface
Finding the intersection of a surface with one of the three coordinate planes (or with a plane parallel to one of the three coordinate planes) helps one visualize the surface.
Such and intersection is called a trace of the surface.
The xy­trace of a surface consists of all points that are common to both the surface and the xy­plane. Similarly the xz­trace of a surface consists of all points that are common to both the surface and the xz­plane
5
PreCal 10.1 The 3D Coordinate System ­ End.notebook
March 29, 2017
Ex. 6 Sketch the xy­trace of the sphere.
(x − 3)2 + (y − 2)2 + (z +4)2 = 52
z
y
x
6