VECTOR ANALYSIS1
VECTOR ANALYSIS1
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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
(i) Addition
(ii) Subtraction
(iii) Multiplication
i. ADDITION OF VECTORS
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
LAWS OF VECTORS – ADDITION
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
(iii) For….. of additive identity
For every vector a, we have;

Where;
0 The null (zero) vector
(iv) Entrance of addictive reverse
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 a^{2} + b^{2} = c^{2}
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 coordinate 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 coordinate 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

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. 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
. =
=
= +
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