Vector Algebra
Vector Algebra
Vector Algebra
VECTOR ALGEBRA
INTRODUCTION:
Electromagnetic Field (EMF) Theory is often called Electromagnetics. It is a
subject which deals with electric field, magnetic field and also electromagnetic fields and
phenomena.
EMF Theory is essential to analyze and design all communication and radar
systems. Infact, it is also used in Bio-Systems and in this context it is called Bio-
Electromagnetics. Source of electromagnetic field is electric charges: either at rest or in
motion. However an electromagnetic field may cause a redistribution of charges that in
turn change the field and hence the separation of cause and effect is not always visible.
The analysis and design of a system, device or circuit requires the use of some
theory or the other.
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The analysis of a system is universally defined as one by which the output is
obtained from the given input and system details.
The design of a system is one by which the system details are obtained from the
given input and output.
These two important tasks of analysis and design are executed by two most
popular theories, namely, CIRCUIT and ELECTROMAGNETIC theories.
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VECTOR ANALYSIS:
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UNIT VECTOR:
A vector A has both magnitude and direction. The magnitude of A is a scalar
written as A or |A|. A Unit Vector, a A along A is defined as a vector whose magnitude is
unity and its direction is along A.
aA = A/|A| = Ā/A
We have,
|a A | = 1. Thus, A can be written as
A = A aA
which completely specifies A in terms of its magnitude A and its direction a A .
A = Ā = √(Ax² + Ay ² + Az²)
and the unit vector along A or Ā is given by
aA = (Ax ax + Ay ay + Az az)/ √(Ax² + Ay ² + Az²)
Properties of Unit Vectors:
i. ax . ax = ay . ay = az . a z = 1
ii. ax x ax = ay x a y = az x a z = 0
iii. ax x ay = a z
iv. ax . ay = 0
v. ay x ax = - az
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Vector Algebra:
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Position and Distance Vector:
A point P in Cartesian coordinates may be represented by (x, y, z).
“The position vector, rp (or radius vector) of point P is as the directed sitance
from the origin O to P.”
rp = OP = x a x + y a y + z a z.
2. Vector Multiplication:
The product of two vectors A and
B is either a scalar or vector depending on
the manner how they are multiplied. There are two
types of vector multiplication.
Scalar or Dot Product: A. B.
Vector or Cross Product: A X B.
Dot Product:
The dot product of two vectors A and B
written as A. B is defined geometrically as the
product of magnitudes of A and B and the cosine
of the smaller angle between them.
A.B = |A| |B| cos θAB
The dot product of two vectors yields a scalar. If A = (Ax,, Ay, Az) and B = (Bx,By, Bz) , then
A.B = Ax Bx + Ay By + Az Bz
which is obtained by multiplying A and B component by component.
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Thus,
A.A = | A| | A| cos 0° = A2
The scalar product of a vector and a unit vector yields the component of the
vector in the unit vector direction. Thus, we have
Ax .A = | Ax| | A| cos α = Ax
Ay .A = | Ay| | A| cos β = Ay
Az .A = | Az| | A| cos γ = Az
Where cos α, cos β and cos γ are the direction cosines with α the angle between A and x-
axis, β the angle between A and y-axis and γ the angle between A and z-axis.
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where dL = vector incremental length (magnitude dL in the direction of path).
The product of a force F and a distance dr represents an incremental amount of
work dW done by the force F in moving an object a distance cos θ dL = dr.
Thus,
dW = F.dL = F cos θ dL
If the path is broken up into segments parallel and perpendicular to F, we note from
above equation that contributions to the work occur only for the segments parallel to F (θ
= 0°) with no work for the segments perpendicular to F (θ = 90°). Summing up the
contributions of the segments parallel to F, we obtain the total work W between the end
points of the point. For finite length segments dL, this value is approximate.
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2√2
= - (1/r)|
√2
= 1/(2√2)
2) Instead of integrating along the direct path from r = √2 (at (x, y) = (1, 1)) to r =
2√2 (at (x, y) = (2, 2)), let us follow a rectangular coordinate path from x, y = 1, 1 to x, y
= 2, 1 (constant y) and then from x, y = 2, 1 to x, y = 2, 2 (constant x).
(F.dL)y = a r/(x2 + y2 ) . a x dx
Since x = y and dx = dy, the x-constant and y-constant terms are equal. Thus, the total
work is twice the work for y-constant path.
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i.e. W = ∫F.dL = 2∫ x/((x2 + y2 ) √(x2 + y2 )) dx
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2 2 2
2 (3/2) 2
=2∫ x/(2x ) dx = (2/√8) ∫ (1/x )dx = (-1/√2) (1/x) |
1 1 1
= 1/(2√2)
Note:
For a vector like F, the line integral depends only on the end points so we could follow
any path. Further, if we integrate F around a closed path, starting say at x, y = 1, 1 and
ending back at x, y = 1, 1, the result is zero.
Thus,
∮ F. dL = 0
where ∮ indicates integration around a closed path. Any field for which the line integral
around a closed path is zero is called a conservative or lamellar field. Not all fields are
lamellar.
Proble ms:
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Where,
ms = mass of satellite, kg.
me = mass of earth = 6 x 1024 kg.
r = distance of satellite from centre of earth, m
G = gravitational constant = 6.67 x 10-11 Nm2 kg-2
The circumference of earth = 40, 000 km.
Cross Product:
The cross product of two vectors A and B, written as A X B, is a vector
quantity whose magnitude is the area of the parallelepiped formed by A and B and is in
the direction of advance of a right – handed screw as A is turned into B.
Thus,
A X B = AB sinθAB a n
Where,
an is a unit vector normal to the plane containing A and B.
θAB is the smaller angle between the two vectors.
The direction of a n is taken as the direction of the right thumb when the fingers of the
right hand rotate from A to B.
AXB=
Reversing the order of the vectors A and B results in a unit vector in the opposite
direction, and we see that the cross product is not commutative, for B X A = - (A X B).
If the definition of cross product is applied to the unit vectors a x and a y, we find
ax X ay = az
for each vector has unit magnitude, the two vectors are perpendicular, and the rotation of
ax into ay indicates the positive z – direction by the definition of a right – handed
coordinate system. In a similar way,
ay X az = ax
az X ax = ay
Cross Product has the following basic properties:
1. It is not commutative:
AXB≠BXA
It is anti – commutative:
A X B = - (B X A)
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2. It is not associative:
A X (B X C) ≠ (A X B) X C
3. It is distributive:
AX (B + C) = A X B + A X C
4. A X A = 0
A simple example of the use of the cross product may be taken from geometry or
trigonometry. To find the area of the parallelogram, the product of the lengths of the two
adjacent sides is multiplied by the sine of the angle between them. Using vector notation
for the two sides, we then may express the (scalar) area as the magnitude of A X B or
|A X B|.
Proble ms:
i.e. A.(B X C) =
Since the result of this vector multiplication is scalar, it is called scalar triple product.
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It should be noted that
(A . B) C ≠ A (B.C)
but
(A . B) C = C (A.B)
Practice Exercise:
1. Find the vector A directed from (2, -4, 1) to (0, -2, 0) in Cartesian coordinates and
find the unit vector along A.
2. Given the vector field, F = 0.4(y – 2x) a x – (200/(x2 +y2 +z2 )) a z.
a) Evaluate |F| at P(-4, 3, 5)
b) Find a unit vector specifying the direction of F at P.
Describe the locus of all points for which
c) Fx = 1
d) |Fz| = 2
3. Let E = 3 a y + 4 a z and F = 4 a x - 10 a y + 5 a z.
a) Find the component of E along F
b) Determine a unit vector perpendicular to both E and F.
4. Show that a = (4, 0, -1), b = (1, 3, 4) and c = (-5, -3, -3) form the sides of a
triangle. Is this a right – angled triangle? Calculate the area of the triangle.
5. Given points A(2, 5, -1), B(3, -2, 4) and C(-2, 3, 1), find
a) RAB, R AC.
b) The angle between RAB and RAC.
c) The length of projection of RAB on RAC.
d) The vector projection of RAB on RAC
6. A triangle is defined by the three points A (2, -5, 1), B (-3, 2, 4) and C (0, 3, 1).
Find
a) RBC X RBA
b) The area of the triangle
c) A unit vector perpendicular to the plane in which the triangle is located.
7. Show that A = 4 a x - 2 a y - a z and B = a x + 4 a y - 4a z are perpendicular.
8. Given A = a x + a y, B = a x + 2a z, C = 2a y + a z, find (A X B) X C and compare it
with A X (B X C).
9. Using the vectors given above find A.BXC and compare it with A X B.C
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