Advanced Math A3
付乳燕~ Fu Ruyan
E-mail: furuyan@cumt.edu.cn
Advanced Math
A3: 48 periods
Mon(3.4), Wed(3.4), Thu(5.6)
12. Vectors and the Geometry of Space
14. Partial Derivatives
15. Multiple Integrals
Reference Books:
1. Weir , Hass and Giordano, Thomas’ Calculus (Eleventh Edition) .
2. James Steward, Calculus (Seventh Edition).
About homework:
We assign homework after each class. Please hand in
it on every Monday.
Examining Method:
Attendance, speak, homework and other performance
in class account for 30%, and the final exam accounts
for70% in the hundred-mark system.
Chapter 12 Vectors and the Geometry of Space
§12.1 Three-Dimensional Coordinate Systems
§12.2 Vectors
§12.3 The Dot Product
§12.4 The Cross Product
§12.5 Lines and planes in Space
§12.1 Three-Dimensional Coordinate Systems
1. Three-dimensional rectangular coordinate system
2. Distance and Spheres in Space
y
First, we recall two-dimensional
rectangular coordinate system.
Any point in the plane can be b P ( a, b)
represented as an ordered pair (a,b)
of real numbers.
x
o a
Point P ( a, b)
1. Three-dimensional rectangular coordinate system
The x- and y-axes are horizontal and the z-axis is
vertical. (Fig 1)
The direction of z-axis is determined by the right-hand
rule. (Fig 2)
The three coordinate axes determine the three coordinate
planes: the xy-plane, the yz-plane and the xz-plane.
These three coordinate planes divide space into eight parts,
called octants.
Ⅲ z
zox plane
yoz plane
Ⅱ
Ⅳ
y Ⅰ
o
xoy plane
Ⅶ x Ⅵ
Ⅴ
Ⅷ
Eight octants
For any point P in space, we represent it by an ordered
triple (a,b,c) of real numbers,where a, b, and c are called
the coordinates of P.
11
P (a, b, c)
z
O(0,0,0)
c C
A(a,0,0)
P
B
y B(0, b,0)
o
A b
x a C (0,0, c)
10
The Cartesian product R R R {( x, y, z ) x, y, z R}
is the set of all ordered triples of real numbers and
is denoted by R 3, which is also the set of all
ponits in space.
z
Three-dimensional
rectangular coordinate
system
o
y
x
In R 2 , the graph of an equation involving x and y
is a curve. For example,
x 1 a line; x y 1 a circle;
2 2
In R 3 , an equation in x, y, and z represents a surface.
Example 1 What surfaces in R 3 are represented by
the following equations?
(1) z 3 (2) y5
Solution
(a) z 3, a palne in R 3 (b) y 5, a palne in R 3
Qu1: what is the graph of equation y 5 in R 2 ？
EXAMPLE 2 Graphing Equations.
What points P(x, y, z) satisfy the equations
x2 y2 4 and z 3
Solution
the circle x 2 y 2 4
in the plane z 3
Remark:
The equation x 2 y 2 4
represents a cylinder
2. Distance and Spheres in Space
The Distance Between P1 ( x1 , y1 , z1 ) and P2 ( x2 , y2 , z 2 ) is
P1 P2 ( x2 x1 ) 2 ( y2 y1 ) 2 ( z 2 z1 ) 2
EXAMPLE 3 Finding the Distance Between Two Points
The distance between P1(2,5,3) and P2(-1,1,3) is
P1 P2 (1 2) 2 (1 5) 2 (3 3) 2
9 16
5
We can use the distance formula to write equations
for spheres in space.
The Standard Equation for the Sphere of Radius a and
Center ( x0 , y0 , z0 )
( x x0 ) 2 ( y y0 ) 2 ( z z0 ) 2 a 2 .
In particular, if the cernter of the sphere is the origin O
and the radius is r, the eqution is
x y z r
2 2 2 2
EXAMPLE 4: Finding the Center and Radius of a Sphere
x2 y2 z 2 4x 6 y 2z 0
Solution
( x 2 4 x 4) ( y 2 6 y 9) ( z 2 2 z 1) 4 9 1
( x 2) 2 ( y 3) 2 ( z 1) 2 14
Center (2,3,1); Radius 14
Summary
• Three-dimensional rectangular system
• The formula of distance in space
• The equation of a sphere in space
Homework
1. Page 6~ exercise 10;
2. Page 7~ exercise 15.16
