微积分教学资料——chapter12

points P in space and ordered triples (, b, c)in R3
z
R(0,0, c)
B(0,b, c)
C(a, o, c)
r
o
x P(a,0,0)
M (a,b,c)
y
Q(0, b,0)
A(a, b,0)
We call a,b and c the coordinates of P
by the symbol
a
or
a
.
a
a2 1
a22
an2
a is called a unit vector
when
a
1
.
If a 0 ,thenthe unit vector that has the same
direction as
a is
u
a a
Definition If
a a1, a2 , , an ,
The zero vertor is denoted by 0
Definition of Vector Addition If u and v are vectors positioned so the initial point of v is at the terminal point of u,then the sum u+v is the vector from the initial point of u to the terminal point of v.
Chapter 12 Vectors and Geometry of Space
12.1 Three-Dimensional Coordinate Systems *12.2 Vectors *12.3 The Dot Product *12.4 The Cross Product * 12.5 Equations of Lines and Planes
op
P(a1, a2 )
a op P(a1, a2 , a3 )
o
o
The three-dimensional vector a a1, a2 , a3 is the
position vector of the point P(a1, a2 , a3 ) .
An n-dimensional vector is an ordered n-tuple: a a1, a2 , , an
a
b
a
b
cos
0
Proof By the Law of Cosines,we have
2 a b
a 2
2 b
2
a
b
cos
Corollary If
vectors a and
is the angle b then
between
the
nonzero
cos
a
b
a
b
Example Find the angle between the vectors
12.6 Cylinders and Quadric Surfaces *12.7 Cylindrical and Spherical Coordinates
In this chapter we introduce vectors and coordinate systems for three-dimensional space. This will be the setting for our study of the calculus of functions of two variables in Chapter 14 because the graph of such a function is a surface in space. In this chapter we will see that vectors provide particularly simple descriptions of lines and planes in space.
Systems z
• origin O
z
o
x
Coordinate axes
y
yz plane
oxy plane y
x
Coordinate planes
Three-Dimensional Rectangular Coordinate Systems
• octants
Ⅲ Ⅳ
Ⅶ
x
Ⅷ
z
Ⅱ
yz plane
of a a
nonzero vector makes with the
a are the angles positive x, y,
and z-axes.
a a1, a2 , a3
, , [0, ]
o
cos,cos , and cos are called the directio cosin of a
Distance Formula in Three Dimensions
The distance P1P2 between the points P1(x1, y1, z1)andP2(x2, y2, z2)
is
P1P2 (x2 x1)2 ( y2 y1)2 (z2 z1)2
z
z2
P1
k j
io
a a1, a2 , a3
k j
io
We have a a1, a2 , a3 a1i a2 j a3k
Definition If
a a1, a2 , , an
and
b b1,b2 , ,bn
,then
the
dot
product
of
a
and
b
is
the
number a
We have
The
scalar
projection
of
b
onto a:
comp ab
b cos
b
a
b
a
b
a a
b
The vector
projection of
b
onto a:
proja b
(
a a
b)
a a
a a
b
2
a
Example Find the projection of b
ps is called the vector is denoted by projab.
b
The number onto a(also
b cos is called the
called the componen
scalar
of
b
projection of along a )and
is denoted by comp ab.
12.1 Three-Dimensional Coordinate Systems
Through point O , three axes vertical each other, by right-hand rule, we obtain a Three-Dimensional Rectangular Coordinate
where a1, a2 , the components
, an of
are real a . We
numbers that are called
denote by v n the set of
all n-dimensional vectors.
The magnitued or length of the vector a is denoted
z1A
x1
O
x2
x
P2
y1
B
C
y2
y
Equation of a Sphere An equation of a sphere with center (h,k,l) and radius r is
(x h)2 ( y k)2 (z l)2 . r 2
In particular, if the center is the origin O, then an equation of the sphere is
then
1.a+b=b+c
2.a+(b+c)=(a+b)+c
3.a+0=a
4.a+(-a)=0
5.c(a+b)=ca+cb 6.(c+d)a=ca+cd
7.(cd)a=c(da)
8.1a=a
The standard basis vectors in v 3
i 1,0,0, j 0,1,0, k 0,0,1
a
2i
j
k
and
b 3i 2 j k
Corollary
a And b are orthogonal if and only if
a
b
0
ɔ:'θɔgənl
Direction Angles and Direction Cosins
The direction
, and
angles , that
We have
cos a1 , cos a2 , cos a3
a
a
a
Tthheedvireecctotironcoofsa,cos,cos is a unit vector in
Projections
b
s
a
p
b
a
sp
Tprhoejevcetciotonrowf itbh
roepnrtoeseantaatinodn
b b1,b2 , ,bn
c is scalar,then
a
b
百度文库
a1, a2 ,
, an
b1,b2 ,
, bn
a1 b1, a2 b2 , , an bn
ca
ca1, a2 ,
,an
ca1, ca2 ,
, can
Properties of Vectors
If a,b.and c are vectors in v n and c and d are scalars,
then
1. a a a 2 2. a b b a 3. a (b c) a b a c 4. (ca) b c(a b) a (cb) 5. 0 a 0
Theorem If is the angle between the vectors a
and b then
x2 y2 z2 r2
z
o
y
x
12.2 Vectors
The term vector is used by scientists to indicate a quantity that has both magnitude and direction.
B Suppose a particle moves along a line
o xy plane
Ⅰ
y
Ⅵ Ⅴ
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 R3.
We have given a one-to-one correspondence between
The diffrence u-v of the two vectors u and v. u-v
v
u
Components
The two-dimensional vector a a1, a2 is the position vector of the point P(a1, a2 ) .
a
u+v v u
The Triangle Law
u+v v
u
The Parallelogram Law
Definition of Scalar Multiplication If c is a scalar and v is a vector,then the scalar multiple cv is the vector whose length is c times the length of v and whose direction is the same as v if c 0and is opposite to v if c 0. If c 0 or v=0,then cv=0.
b given
by
a b a1b1 a2b2 anbn
The dot product is also called the scalar product (or inner product).
Properties of the Dot Product
If a,b.and c are vectors in v n and c is scalar,
v
segment from A to point B.
A
Initial point(the tail) A
Terminal point(the tip) B The displacement vector is denoted by v =AB=v
B v A
D u C
u and v are equivalent u=v