# FORM SIX MATHEMATICS STUDY NOTES VECTOR ANALYSIS-1

FORM SIX MATHEMATICS STUDY NOTES VECTOR ANALYSIS-1

UNAWEZA JIPATIA NOTES ZETU KWA KUCHANGIA KIASI KIDOGO KABISA:PIGA SIMU:0787237719

## FORM SIX MATHEMATICS STUDY NOTES VECTOR ANALYSIS-1

Definition
Vectors are any quantity that possess both magnitude and direction .

Example

(i)  Displacement

(ii)  Velocity

(iii)  Acceleration

REPRESENTATION OF A VECTOR

The vector quantity is always described by using two capital letters with respect to arrow on top or small letters with respect to bar at the bottom.

n

A                                         B

or

COMPONENTS OF VECTORS.

This depends on the dimension of vector as follows;

(i)     For two dimensions namely as  and

Where  =x- value

= y – value

e.g.     = (x,y) coordinate form

component form

Diagram

For three dimensions

Re:

This involves three components namely as , and

Where,

= x – value

= y – value

TERMINOLOGIES APPLIED IN VECTOR ANALYSIS

1.  PARALLEL VECTORS

These are vectors having the same direction.
e.g

2.      EQUAL VECTORS

These are vectors having the same magnitude and direction.

e.g.

= 5N

= 5N

3.      NEGATIVE (OPPOSITE) VECTORS

These are vectors having the same magnitude but opposite direction.

Hence

(i)     =- (vector in opposite direction

(ii)     = = b vector on the same direction

FREE VECTORS

These are vectors which originate from different points.

POSITION VECTORS

These are vectors which originate from the same points.

NB:

–          Position vector of   =  –

–          Position vector of  = –

–          Position vector of  =    –

–          Position vector of  =    –

NULL VECTORS

These are vectors which have a magnitude of zero ( length of zero) or

These  are vectors which contain zero point

Eg.

= (0,0)

=  (0,0,0)

COLLINEAR VECTORS

– These are vectors which lie on the same line.

i.e.

COPLANAR VECTORS

These are vectors which lie on the same plane.

,  and  are coplanar vector

NB:

Consider the vector diagram below;

Where

= initial (starting) point

B = Final (terminal) point

OPERATION IN VECTORS

These are

(ii)  Subtraction

(iii)  Multiplication

Suppose two dimensional vectors

Suppose three dimensional vectors

+

k

RESULTANT VECTOR

Is the single vector   which represents the effect of all vectors acting at a point.

Consider  are acting at a point

Hence

Where F is the result force/ vector

A: TRIANGULAR LAW OF VECTORS APPLICATION

Consider the vector diagram below;

+  –  = 0

=  +

 =   +

Where r is the resultant vector

B. PARALLELOGRAM LAW OF VECTORS ADDITION

–         Consider the vector  diagram below

+  = …………………………(i)

+   –  = 0

+  =  ……………………………(ii)

From (i) and (ii) above

Proved

(ii) Addition of vectors is associative for any three vectors a, b and c

Proof

Consider the vector diagram below;

Individual but not considered

=

=

=

……(i)

…………………..(ii)

From (i) and (ii) above

=

Proved

For every vector a, we have;

Where;

0                 The null (zero) vector

For every vector a we have

Where

→ is the positive of vector

→ is the null vector

ii. SUBTRACTION OF VECTORS

Suppose two dimensional vectors

Hence

=

=

–          Suppose three dimensional vectors

Hence

Question 1

1.      If

(a         Find   (i)

(ii)

Comment of the results in (a) above

Question 2

Given that

(i)     Find

(ii)

(b) Comment on the results in a above

MAGNITUDE OF A VECTOR

– The magnitude of a vector is a measure of length of the vector.

–     – This is denoted by the symbol

(a)   Consider two dimensional vector

By using Pythagoras theorem

Recall;

Where

– is the magnitude/ module of the vector r

(b)  Consider three dimensional vector

RECTANGULAR RESOLUTION OF A VECTOR

Let:   be three rectangular axes and  be three unit vectors parallel to  axes respectively.

Consider

+

+

Also consider the right angled  OFP

Using Pythagoras theorem

i.e a2 + b2 = c2

Where

= is the magnitude of the vector

Question

Given that

Find

DIRECTION RATIO AND DIRECTION COSINES

I.   DIRECTION RATIO

Suppose the vector

The direction ratio is given by

II.     DIRECTION COSINE

Consider the vector

From three dimension plane.

Makes angles  with  direction respectively

Hence

Therefore the direction cosines are

FACT IN DIRECTION COSINES

– Suppose the vector

Also the direction cosines are

Hence

++= 1

– The sum of square  of the direction cosines is one.

Proof

i.e  = x+y+z

=

Also

—————-(i)

———————-(ii)

—————————-(iii)

Adding the equation (i) , (ii) and (iii)

+=  +

=

But

+=

 +

UNIT VECTOR
-Is the vector whose magnitude (modules) is one line a unit
-The unit vector in the direction  of vector a is donated  by  read as “ a cap” thus
NOTE:
Any vector can be compressed as the product of it’s magnitude and it’s unit vector
i.e

QUESTIONS

1.  Find a vector in the direction of vector which has a magnitude of 8 units

2.  Find the direction ratio and direction cosines of the vector   where p is the point (2, 3, -6)

THE FORMULA OF DISTANCE BETWEEN TWO POINTS

Suppose the line joining the points  and  whose position  vectors are a and b respectively

HENCE

=

=

Hence

-Formula  distance between two point

MID POINT OF A LINE

Suppose M is the point which divide the line joining the points and  whose position vectors are respectively a and b into two equal parts

i.e

a =

b =

Hence

Therefore

The co-ordinate of M is

INTERNAL AND EXTERNAL DIVISION OF A LINE (RATIO THEOREM)

I.     INTERNAL DIVISION OF A LINE

–          Suppose M- is the point which divides the line joining the points   and  whose point vectors are a and b respectively internally in the ratio X:ee

a = =

b = =

……………(i)

……………(ii)

By using ratio theorem

By using multiplication

The ordinate form of M is

II.      EXTERNAL DIVISION OF A LINE

Suppose M- is the point which divides the line joining the points   and  where position vectors are   respectively , externally is the ration

=

……………(i)

……………(ii)

By using ratio theorem

i.e

BY CROSSING MULTIPLICATION

Therefore

The co-ordinate of M

External division of a line where

QUESTIONS

1.Find the length of the line  of

2.  Find the position vector which divides line  having point  into two equal points.

3.  A  and B are two points whose vectors are 3 +  and    respectively. Find the position vector of the points dividing AB.

(a) Internally in the ratio 1:3

(b) Externally in the ratio 3:1

III.    MULTIPLICATION OF A VECTOR

(A)    SCALAR MULTIPLICATION OF A VECTOR

In this case a vector is multiplied by a certain constant called scalar

Let

THEREFORE

QUESTIONS

If

(a)   Find (i)

(ii).

(b)  Comment on results in  above

DEFINITION OF DOT PRODUCT

For vectors

Where

is the above between

cos Q

 cos Q

CHARACTERISTICS

1.      1. PARALLEL VECTOR

Two vector are said to be parallel of the angle between them is zero

Mathematically

From

= cos Q

But Q =

= cos Q

This is one among the

2. Orthogonal vectors

Two vectors are said to be  orthogonal of the angle between them is 90

Mathematically

From

But Q = 90 (orthogonal or perpendicular vector)

= cos 90

This is conditional for the orthogonal vector

THEOREM.

(a)For the definition

cos

=(1,0)

=(1,0)

Therefore

From the definition

cos

cos 90

x 0

Therefore

–      Suppose the vector

=

= +  ++

=

From the definition

= cos

= cos

–      Consider the vector

QUESTIONS

1.      If + 2+2
Find the angle between  and

2.      Show that the vectors
=
are orthogonal

3.      If =2, =3
, find

4.      The vectors      and
where k are such that
and   are orthogonal find k

5.      If      and
=
Find the value of  if  and  are orthogonal

6.If =2, =3,
Find

APPLICATION OF DOT PRODUCT

1.   TO VERIFY COSINE RULE

Consider the vector diagram below

+=

…………………..(i)

Dot by

. =

=

=  +