Vectors - University of Babylon

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


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


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


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


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


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


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


( ) ( ) ( )

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


4. |

|

5. The unit vectors i, j and k have the following multiplication

table:

Products of Three Vectors:

( ) |

|

Chapter Two Vectors


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


( ) ( ) ( )

( ) [( ) ( ) ( ) ] ( )

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


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


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


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


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

( ) ( )

( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( ) ( )

Chapter Two Vectors


Example:

Find the derivative of

( )

at the point ( )

( )

( ) corresponds to

( )

Vectors - University of Babylon

Documents

Transcript of Vectors - University of Babylon