Lecture 2.
Vectors. The elementary operations over vectors. The scalar, vector and mixed product of vectors.
§1. The elementary operations over vectors
A vector is a matrix that has only one row – then we call the matrix a row vector – or only one column – then we call it a column vector.
A
row vector is of the form:
.
A
column vector is of the form:
.
A quantity such as force, displacement, or velocity is called a vector and is represented by a directed line segment
A vector
in the plane is directed line segment. The directed line segment
has initial point
A and terminal
point
B; its length
is denoted by
.
Two vectors are equal
if they have the same length and direction.
Component form
If
is a two dimensional
vector in the plane equal to the vector with initial point at the
origin and terminal point
,then the
Component form
of
is:
If
is a three dimensional
vector in the plane equal to the vector with initial point at the
origin and terminal point
,
then the
Component form
of
is:
The numbers
are called the components of
.
Given the points
and
,
the standard position vector
equal to
is
The magnitude
or length
of the vector
is the nonnegative number
The only vector with length 0
is the zero vector
or
.
This vector is also the only vector with no specific direction.
Vector Addition and Multiplication of a vector by a scalar
Let
and
be vectors with k
a scalar.
Addition:
Scalar
multiplication:
If the length of
is the absolute value of the scalar
times the length of
.
The vector
has the same length as
but points in the opposite direction.
If
and
,
Note
that
and the difference
as the sum
Properties of vector operations:
Let
be vectors and
be scalars.
1)
2)
3)
4)
5)
6)
7)
8)
9)
Unit vectors
A vector of length 1 is called unit vector. The standard unit vectors are:
We call the scalar (or number)
the i-component
of the vector
,
the
j-component
of the vector
,
and
the k-component.
In component form,
and
is
.
If
,
then
is a unit vector in the direction of
,
called the direction
of the nonzero vector
.
2 The inner, vector and mixed product of vectors
Inner Product and its Properties
Definition 1.
The inner product of two vectors
and
is the product of the absolute values of these vectors and the cosine
of the angle between them:
.
Property 1. The inner product of two vectors is equal to the product of the absolute value of one vector and the projection of the second vector onto the first, i.e.,
.
Property 2. The inner product of two vectors is equal to zero if and only if these vectors are perpendicular.
1. If
,
then
,
and the inner product equals
.
2. Conversely, suppose that
Property 3. The inner product of vectors is commutative:
Property 4. To multiply an inner product by a number , it is sufficient to multiply one of the factors by :
(Without proof.)
Property 5. Inner product is associative:
.
and
.
The last property allows us to multiply these vectors term by term:
.
Since the unit vectors are mutually perpendicular
(
),
their inner products are zero (by Property 2).
Consider the product of a unit vector with itself:
;
;
.
Thus, six of the nine terms are zero, and the remaining three terms are
.
The inner product of vectors is equal to the sum of products of their coordinates.