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Chapter Two Vectors
Advanced Calculus II Hawraa Abbas Almurieb
Vectors
1. Vectors in Two Dimensions
In order to distinguish vectors from scalar, we will se hod-faced
letters, to denote vectors; for example, a, b, U, and so forth. Vectors
have both, magnitude and direction, while scalars have only
magnitude. For that, we will represent vectors by arrows. The
magnitude of the vector a is the length of the arrow, and its
direction is the direction of the arrow.
When the vectors intial point is placed at the origin, we write:
( )
to denote the vector from O to R.
Length of a:
The magnitude of a is the distance from O to R, and is denoted by
‖ ‖:
‖ ‖ √
terminal
point
Chapter Two Vectors
Advanced Calculus II Hawraa Abbas Almurieb
Equality:
The vectors ( ) and ( ) are equal if and only if
Zero:
The zero vector is the vector of length zero. As a result, if its intial
point is at the origin, then so its terminal point, so that
( )
Addition:
If the intial point of b is placed at the terminal point of a, then a+b is
the vector drawn from the intial point of a to the terminal point of b.
( )
Multiplication of a vector by a scalar:
Let a be a vector, and be a scalar (real number). If then
or is the vector whose direction is the same as that of a and
whose length id times the length of a.
Chapter Two Vectors
Advanced Calculus II Hawraa Abbas Almurieb
If then the direction of is opposite to that of a and whose
length is | | times the length of a.
If , or a=0, then is the zero vector.
( )
Subtraction:
The vector is drawn from the terminal point of b to the
terminal point of a.
( )
Chapter Two Vectors
Advanced Calculus II Hawraa Abbas Almurieb
Finding Vectors using their end points:
The vector whose intial point is ( ) and terminal point
is ( ) denoted by , can be found by subtracting the
coordinates of its intial point from the corresponding coordinates of
its terminal point:
( ) ( ) ( )
The Dot Product:
we have two ways to multiply a vector by another. The first type is
dot product which gives a scalar as a result of multiply two vectors
together.
Relation between dot product and length of a vector:
When we multiply a vector by itself using dot product, we get
‖ ‖
Also, if is a scalar, then
‖ ‖ ‖ ‖ √
| |√
| |‖ ‖
Chapter Two Vectors
Advanced Calculus II Hawraa Abbas Almurieb
Unit Vectors:
Any vector of length one is called a unit vector. To determine a unit
vector of length one in the direction of a vector a we use:
‖ ‖
Geometrical Interpolation of Dot Product:
Let a and b be nonzero vectors which have the same initial point.
And let be the angle between them s.t. . Use the law of
cosines from trigonometry for the triangle below:
‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖‖ ‖
We obtain
‖ ‖‖ ‖
We can use the relation above to find the angle between vectors as
follow:
‖ ‖‖ ‖
‖ ‖
‖ ‖
Chapter Two Vectors
Advanced Calculus II Hawraa Abbas Almurieb
where and are unit vectors of and respectively.
Orthogonal Vectors:
The vectors and are orthogonal if and only if
Unit Coordinate Vectors:
Unit vectors in the direction of x-axis and y-axis are important
enough so that the special symbols and are reversed for them.
Their components are
( ) ( )
Also, it is readily to verify that
Q: Prove that i and j are orthogonal?
We can express a given vector ( ) in terms of i and j, we
have
( )
Triangle Inequality:
‖ ‖ ‖ ‖ ‖ ‖
Chapter Two Vectors
Advanced Calculus II Hawraa Abbas Almurieb
2. Vectors in Three Dimensions
A three-dimensional vector a may be thought of as an arrow in
three-dimensional -space. Let us place the initial point at the
origin, then the terminal point coincides with some point P. If
( ), then we write
( ),
The length of a: ‖ ‖ √
Equality: The vectors if and only if
Zero: The zero vector is ( )
Addition: ( )
Multiplication of a vector by a scalar: ( )
Subtraction: ( )
Dot Product:
Unit Vectors:
‖ ‖
There are now three unit coordinate vectors, denoted by
Chapter Two Vectors
Advanced Calculus II Hawraa Abbas Almurieb
( ) ( ) ( )
In terms of these vectors, we can write the vector ( ) as
,
Cross Product:
In dot Product, we get a scalar as a result of multiplication. In cross
product, we multiply to vectors and get another vector as a result.
Let and be nonzero vectors, and be the angle between them,
s.t. . Then is defined to be the vector with the
following properties:
1. is perpendicular at to both a and b
2. The magnitude of is given by
‖ ‖ ‖ ‖‖ ‖
3. The direction of is chosen so that when a is rotated into b
through the angle , then a, b, form a right handed
system of vectors.
Chapter Two Vectors
Advanced Calculus II Hawraa Abbas Almurieb
4. |
|
5. The unit vectors i, j and k have the following multiplication
table:
Products of Three Vectors:
( ) |
|
Chapter Two Vectors
Advanced Calculus II Hawraa Abbas Almurieb
3. Planes and Lines
The algebra of vectors can be used to study the properties of planes
and straight lines in three dimensional spaces.
Planes:
Let ( ) be a given point, and let a given vector n determine
a direction at . Then the plane that contains and is
perpendicular to n consists of all those points ( ) such that the
vector from to is perpendicular to n. the vector n is called a
normal vector to the plane.
Let and be the position vectors of and , respectively; then
is the vector from to . The vectors and n must be
perpendicular, so
( ) (1)
Equation (1) is the basic vector equation of the plane through
that is perpendicular to n.
Example:
Find an equation of the plane that is perpendicular to the vector
, and containing the point ( ).
( ) ( )
Chapter Two Vectors
Advanced Calculus II Hawraa Abbas Almurieb
( ) ( ) ( )
( ) [( ) ( ) ( ) ] ( )
Lines:
A convenient way to describe a straight line is to specify a point on
the line and a vector parallel to the line. Let L be the line through the
point ( ) having the same direction as the a vector
Let be the position vector
of . Let be the position vector of an arbitrary
point P on the line L. Then, the difference between and r is
proportional to a. If we denote the proportionality factor by t, then
or
is a vector parametric equation of the line L.
Chapter Two Vectors
Advanced Calculus II Hawraa Abbas Almurieb
The scalar parametric equations are
The symmetric, or Cartesian equations are
Example:
Find vector parametric, scalar parametric, and symmetric equations
of the line through the point ( ) and parallel to the vector -
.
The vector parametric equation:
( ) ( ) ( ) ( )
The scalar equations:
The symmetric equation:
Chapter Two Vectors
Advanced Calculus II Hawraa Abbas Almurieb
4. Vector Functions of One Variable
Suppose that we are given the vector function F defined by
( ) ( ) ( ) ( ) (1)
where and are scalar functions of variable t. comparing this
with , we get
( ) ( ) ( ) (2)
An example for function variable is the equations of straight line
(3)
Geometrically, suppose that lies in some interval . Then
for each value of t in the interval, Equation (1) determines a vector r
that can be regarded as the position vector of a point P. the
collection of points obtained in this way form an arc or curve.
Chapter Two Vectors
Advanced Calculus II Hawraa Abbas Almurieb
Limit:
The statement
( )
means that as approaches , the vector F(t) approaches the vector
. For each s.t.
if | | then ‖ ( ) ‖
i.e. if
( )
( )
( )
Example:
If ( ) ( ) (
) , find ( )
( )
Continuity:
is continuous at if is defined at and
( ) ( )
Chapter Two Vectors
Advanced Calculus II Hawraa Abbas Almurieb
i.e. if
( ) ( )
( ) ( )
( ) ( )
Thus, is continues at iff and are continues at
Example:
The vector equation ( ) ( ) (
) is continues.
Differentiability:
For a given function , we form the difference quotient
( ) ( )
( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( ) ( )