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Vector quantities have both direction as well as magnitude such as velocity, acceleration,
force and momentum etc. We will use for any general vector and its magnitude by In diagrams vectors are denoted by arrows: the length of the arrow is proportional to the magnitude of the vector, and the arrowhead indicates its direction. Minus()is a vector with the same magnitude as but of opposite direction.
We define fourvector operations: addition and three kinds of multiplication.
Let and be unit vectors parallel to the x, y and z axis, respectively. An arbitrary
vectorcan be expanded in terms of these basis vectors
The numbers A_{x} , A_{y} , and A_{z} are called component of geometrically, they are the projections ofalong the three coordinate axes.
(i) Rule: To add vectors, add like components.
(ii) Rule: To multiply by a scalar, multiply each component.
Because and are mutually perpendicular unit vectors
Accordingly,
(iii) Rule: To calculate the dot product, multiply like components, and add.
In particular,
Similarly,
(iv) Rule: To calculate the cross product, form the determinant whose first row is whose second row is (in component form), and whose third row is
Example 1: Find the angle between the face diagonals of a cube.
The face diagonals and are
Example 2: Find the angle between the body diagonals of a cube.
The body diagonals and are
Example 3: Find the components of the unit vector nˆ perpendicular to the plane shown in the figure.
The vectors and can be defined as
Since the cross product of two vectors is itself a vector, it can be dotted or crossed with a
third vector to form a triple product.
(i) Scalar triple product:
Geometrically is the volume of the parallelepiped generated by and , sinceis the area of the base, and is the altitude. Evidently,
In component form
Note that the dot and cross can be interchanged:
(ii) Vector triple product:
The vector triple product can be simplified by the socalled BACCAB rule:
Position, Separation and Displacement Vectors
The location of a point in three dimensions can be described by listing its Cartesian
coordinates (x,y,z) . The vector to that point from the origin is called the position vector:
Its magnitude, is the distance from the origin,
and is a unit vector pointing radially outward.
Note: In electrodynamics one frequently encounters problems involving two points
typically, a source point, where an electric charge is located, and a field point,, at which we are calculating the electric or magnetic field. We can define separation vector
from the source point to the field point by
Its magnitude is
and a unit vector in the direction from
In Cartesian coordinates,
The infinitesimal displacement vector, from (x,y,z) to ( x + dx, y + dy, z + dz), is
For closed surface area element is perpendicular to the surface pointing outwards as shown in figure below.
(i) For x = 2 plane,
(ii) For x = 0 plane,
(iii) For y = 2 plane,
(iv) For y = 0 plane,
(v) For z = 2 plane,
(vi) For z = 0 plane,
For open surface area element is shown in figure below (use right hand rule)
Volume Element
Volume element dτ = dxdydz
Curvilinear Coordinates
Spherical Polar Coordinates
In spherical polar coordinate any general point P lies on the surface of a sphere. The
spherical polar coordinates r,θ,∅ of a point P are defined in figure shown below; r is
the distance from the origin (the magnitude of the position vector), θ (the angle drawn
from the z axis) is called the polar angle, and ∅ (the angle around from the x axis) is the
azimuthal angle.
Their relation to Cartesian coordinates (x,y,z) can be read from the figure:
x = r sinθ cos∅,y = r sinθ sin∅, z = r cosθ
and
The range of r is 0 → ∞, θ goes from 0 → π , and ∅ goes from0 → 2π .
Figure shows three unit vectors , pointing in the direction of increase of the
corresponding coordinates. They constitute an orthogonal (mutually perpendicular) basis
set (just like), and any vector can be expressed in terms of them in the usual
way:
A_{r} , A_{θ} , and A∅ are the radial, polar and azimuthal components of .
Infinitesimal Displacement Vector
An infinitesimal displacement in the direction is simply dr (figure a), just as an
infinitesimal element of length in the x direction is dx :
dl_{r} = dr
On the other hand, an infinitesimal element of length in the direction (figure b) is r dθ
dl_{θ }= rdθ
Similarly, an infinitesimal element of length in the direction (figure c) is rsin θd∅
dl_{∅}_{ }= r sinθd∅
Thus, the general infinitesimal displacement dl is
This plays the role (in line integrals, for example) that = played in
Cartesian coordinates.
Area Element
If we are integrating over the surface of a sphere, for instance,
then r is constant, whereas θ and ∅ change, so
on the other hand, if the surface lies in the xy plane, then θ is
constant (θ = π/2) while r and ∅ vary, then
Volume Element(d/τ)
The infinitesimal volume element dτ, in spherical coordinates, is the product of the three infinitesimal displacements:
Transformation of Vector to Spherical Polar
We can transform any vector in Cartesian coordinates to Spherical polar coordinate as
Thus
where x = rsin θcos∅, y = rsinθsin∅, z = r cosθ
and use table given below:
The cylindrical coordinates r,∅,z o f a point P are defined in figure. Notice that ∅ has the same meaning as in spherical coordinates, and z is the same as Cartesian; r is the distance to P from the z axis, whereas the spherical coordinate /'is the distance from the origin. The relation to Cartesian coordinates is
x = rcos∅, y = r sin∅, z = z
The range of r is 0 → ∞ , ∅ goes from 0 → 2π , and z from  ∞ to ∞
The infinitesimal displacements are
so
and volume element is dτ = r dr dФ, dz.
We can transform any vector in Cartesian coordinates to cylindrical coordinates as
Thus,
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