VECTOR ANALYSIS-1
VECTOR ANALYSIS-1
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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 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
|
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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