Three-dim coordinate space

1
CHAPTER 4 Vector Spaces
Three-dim coordinate space
2
Three-dim coordinate space
We call the number a , b , c the components of
the vector v
The point P(a,b,c) determines the vector v(a,b,c)
The position vector
3
Addition of vectors
Multiplication of a vector by a scalar
4
The length of a vector
is defined to be the distance of the
point p(a,b,c) from the origin
Find
The geometric interpretation
Cgt0, The vector c v is of length c(length v)
of the same direction Clt0, The vector c v is
of length c(length v) of the opposite direction
5
The geometric interpretation
Triangle law of addition Parallelogram law of
addition
6
Vector Space
Set S with vector addition and multiplication
by scalar is a vector space if these operations
satisfy the following
S is a vector space
7
Addition of vectors
Multiplication of a vector by a scalar
8
Linear combination
By solving the linear system
W1 is a linear combination of u1,u2,u3 W1 is a
linear combination of u1,u2,u3 W1 is a linear
combination of u1,u2,u3
W4 is a linear combination of u1,u2,u3
9
Linear combination
Cab w1 be written as a linear combination of
u1,u2,u3
10
Linearly dependent vectors
are said to be linearly dependent provided that
one of them is a linear combination of the other
two
11
Linearly dependent vectors
are said to be linearly dependent provided that
one of them is a linear combination of the other
two
otherwise, they are linearly independent
TH3 the three vectors u1,u2,u3 are linearly
dependent iff there exist scalars c1,c2,c3 not
all zeros such that
NOTE if the only solution of () is
c10,c20,c30 then u1,u2,u3 are linearly
independent
12
Linearly dependent vectors
NOTE if the only solution of () is
c10,c20,c30 then u1,u2,u3 are linearly
independent
TH4 the three vectors u1,u2,u3 are linearly
independent iff
13
(No Transcript)