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2 Vectors In 2-space
And 3-space
Overview
In this chapter we review the related concepts of
physical vectors, geometric vectors, and algebraic
vectors. To provide maximum geometric insight, we
concentrate on vectors in two-space and three-space.
Later, in Chapter 3, we will generalize many of the
ideas developed in this chapter and apply them to a
study of vectors in n-space, that is, to vectors in Rn. A
major emphasis in Chapter 3 is on certain
fundamental ideas such as subspaces of Rn and the
dimension of a subspace. As we will see in Chapter 3,
concepts such as subspace and dimension are directly
related to the geometrically familiar notions of lines
and planes in three-space.
Core sections
•
•
•
•
Vectors in the plane
Vectors in space
The dot product and the cross product
Lines and planes in space
2.1 Vectors in the plane
1. Three types of vectors
(1)Physical vectors:
A physical quantity having both magnitude and
direction is called a vector.
Typical physical vectors are forces, displacements,
velocities, accelerations.
(2) Geometric vectors:
The directed line segment from point A to point B
is called a geometric vector and is denoted by

  AB.
For a given geometric vector AB ,the endpoint A
is called the initial point and B is the terminal
point.
(3) Equality of geometric vectors
All geometric vectors having the same direction
and magni-tude will be regarded as equal,
regardless of whether or not they have the same
endpoints.
y
B
F
A
D
E
x
C
(4) Position vectors
B
y
P
A
O
OP is the position vector for AB.
x
(5) Components of a vector
Let   OP is the position vector for AB. The coordinates
of P are called the components of  .
If A  (a1 , a 2 ) and B  (b1 , b2 ), then
The x  component of AB is b1  a1
The y  component of AB is b2  a 2 .
(6) An equality test for Geometric Vectors
Theorem2.1.1: Let AB and CD be geometric
vectors. Then AB  CD if and only if their
components are equal.
(7) Algebraic vectors:

Theorem2.1.2: Let   AB be a geometric vector,


with A=(a1,a2) and B=(b1,b2). Then
can be
represented by the algebraic vector
b1  a1 
 

b

a
 2 2


or   {b1  a1 , b2  a2 }.
2. Using algebraic vectors to calculate the sum
of geometric vectors
Theorem2.1.2: Let u and v be geometric vectors
with algebraic representations given by
u1 
u 
u2 
v1 
and v   .
v2 
Then the sum u+v has the following algebraic
representation:
u1  v1 
uv  
.

u2  v2 
3. Scalar multiplication
Theorem2.1.3: Let u be a geometric vectors with
algebraic representations given by
u1 
u   .
u2 
Then the scalar multiple cu has the following
algebraic representation:
cu1 
cu   .
cu2 
4. Subtracting geometric vectors
5. Parallel vectors
Vectors u and v are parallel if there is a nonzero
scalar c such that v=cu. If c>0, we say u and v have
the same direction but if c<0, we say u and v have
the opposite direction.
6. Lengths of vectors and unit vectors
7. The basic vectors i and j
2.1 Exercise P126 26
2.2 Vectors in space
1. Coordinate axes in three space
2. The right-hand rule
3. Rectangular coordinates for points in three
space
axis;coordinate planes;octants
4. The distance formula
Theorem2.2.1: Let P=(x1,y1,z1) and Q=(x2,y2,z2) be two
points in three space. The distance between P and Q,
denoted by d(P,Q), is given by
d ( P, Q )  ( x2  x1 )2  ( y2  y1 )2  ( z2  z1 )2 .
5. The midpoint formula
Theorem2.2.2: Let P=(x1,y1,z1) and Q=(x2,y2,z2) be two
points in three space. Let M denote the midpoint of the line
segment joining P and Q. Then, M is given by
 x1  x2 y1  y2 z1  z2 
M 
,
,
.
2
2 
 2
6. Geometric vectors and their components
7. Addition and scalar multiplication for vectors
8. Parallel vectors, lengths of vectors, and unit
vectors
9. The basic unit vectors in three space
2.3 The dot product and the cross
product
1. The dot product of two vectors
Definition 2.3.1: Let u and v are vectors, then the dot
product of u and v, denoted u·v, is defined by
u · v=||u|| ||v|| cosθ.
where θis the angle of vectors u and v.
Definition 2.3.2: Let u and v are two-dimensional vectors,
then the dot product of u and v, denoted u·v, is defined by
u · v=u1v1+u2v2.
Let u and v are three-dimensional vectors, then the dot
product of u and v, denoted u·v, is defined by
u · v=u1v1+u2v2+ u3v3.
2. The angle between two vectors
u · v=||u|| ||v|| cosθ.
3. Algebraic properties of the dot product
(1) u  u  0
( 2) u  v  v  u
(3) u  (cv)  c(u  v)
(4) u  (v  w)  u  v  u  w
4. Orthogonal Vectors(正交向量）
When θ=π/2 we say that u and v are perpendicular or
orthogonal.
Theorem 2.3.1: Let u and v are vectors, then u and
v are orthogonal if and only if
u · v=0.
In the plane, the basic unit vectors i and j are
orthogonal. In three space, the basic unit vectors i, j
and k are mutually orthogonal.
u
5. Projections
θ
q
v  projq (u )  u cos ( )
q
v
q
6. The cross product
Definition 2.3.3: Let u and v are vectors, then the cross
product of u and v, denoted u×v, is a vector that it is
orthogonal to u and v, and u,v,u×v is right-hand system,
and the norm of the vector is
||u×v||=||u|| ||v|| sinθ.
where θis the angle of vectors u and v.
7. Remember the form of the cross product
(two methods)
i j k
u  v  u1 u2 u3  (u2v3  u3v2 )i  (u1v3  u3v1 ) j  (u1v2  u2v1 )k
v1 v2 v3
u  v  {u2v3  u3v2 , u3v1  u1v3 , u1v2  u2v1}
8. Algebraic properties of the cross product
(1) u  v  v  u
(2) u  u  0
(3) (cu )  v  u  (cv)  c(u  v)
(4) u  (v  w)  u  v  u  w
(5) u  (v  w)  (u  v)  w
9. Geometric properties of the cross product
10. Triple products（三重积）
u  (v  w) and u  (v  w)
u1
u  (v  w)  v1
w1
u2
v2
w2
u3
v3
w3
11. Tests for collinearity and coplanarity
Theorem: Let u, v and w be nonzero three
dimensional vectors.
(a) u and v are collinear if and only if u×v=0.
(b)u, v and w are coplanar if and only if u·(v×w).
2.3 Exercise P148 48
2.4 Lines and planes in space
1. The equation of a line in xy-plane
Let P( x0 , y0 ) is a point and k is slope, then the
equation of the line is y  k ( x  x0 )  y0
y
.
l
P0=(x0,y0)
O
x
2. The equation of a line in three space
(1)Point and directional vector form equation of
a line

Let P ( x0 , y0 , z0 ) is a point and s  {l , m, n}
is a directiona l vector, then the line that cross the

point P and along the vector s is given by
x  x0 y  y0 z  z0


l
m
n
(2)Parametric equations of a line

Let P ( x0 , y0 , z0 ) is a point and s  {l , m, n}
is a directiona l vector, then the line that cross the

point P and along the vector s is given by
 x  x0  lt

 y  y0  mt
 z  z  nt
0

Example1: Let L be the through P0=(2,1,6), having
direction vector u given by u=[4,-1,3]T.
(a) Find parametric equations for the line L.
(b) Does the line L intersect the xy-plane? If so,
what are the coordinates of the point of
intersection?
Example2: Find parametric equations for the line L
passing through P0=(2,5,7) and the point
P1=(4,9,8).
2. The equation of a plane in three space
Point and normal vector form equation
of a plane

Let P( x0 , y0 , z0 ) is a point and n  { A, B, C}
is a normal vector, then the plane that cross the

point P and is vertical to the vector n is given by
A( x  x0 )  B( y  y0 )  C ( z  z0 )  0
Ax  By  Cz  D  0
Example3: Find the equation of the plane
containing the point P0=(1,3,-2) and having
normal n=[5,-2,2]T.
Example4: Find the equation of the plane passing
through the points P0=(1,3,2), P1=(2,0,-1), and
P2=(4,5,1).
The relationship between two lines or two planes
Two lines
L1 :
x  x1 y  y1 z  z1


l1
m1
n1
L2 :
x  x2 y  y 2 z  z 2


l2
m2
n2
1. L1 // L2 
l1 m1 n1


l 2 m2 n 2
2. L1  L2 
l1l 2  m1m2  n1n2  0
3. cos 
A line and a plane
Two plane
x  x0 y  y 0 z  z 0
L:


l
m
n
 : Ax  By  Cz  D  0
 1 : A1 x  B1 y  C1 z  D1  0
 2 : A2 x  B2 y  C 2 z  D2  0
1. L //   Al  Bm  Cn  0 1.  1 //  2 
A B C
2. L     
l
m n
3. sin  
| l1l 2  m1m2  n1n2 |
Al  Bm  Cn
l12  m12  n12 l 22  m22  n22
A2  B 2  C 2 l 2  m 2  n 2
A1
B
C
 1  1
A2 B2 C2
2.  1   2 
A1 A2  B1 B2  C1C 2  0
3. cos 
| A1 A2  B1 B2  C1C 2 |
A12  B12  C12
A22  B22  C 22