Vector Algebra | Mathematical Methods - Physics PDF Download

Vector Operations

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 four-vector operations: addition and three kinds of multiplication.

• Addition of two vectors
  Place the tail of  at the head of ; the sum,+ is the vector from the tail of  to the head of
  Addition is commutative:  
  Addition is associative:
  To subtract a vector, add its opposite:
• Multiplication by scalar
  Multiplication of a vector by a positive scalar a, multiplies the magnitude but leaves the direction unchanged. (If a is negative, the direction is reversed.) Scalar multiplication is distributive:
  
  
  
• Dot product of two vectors
  The dot product of two vectors is define by
  
  
  where θ is the angle they form when placed tail to tail. Note that . is itself a scalar.
  The dot product is commutative,
  
  and distributive,
  Geometrically  is the product of A times the projection of  along  (or the product of B times the projection of )
  If the two vectors are parallel,
  If two vectors are perpendicular, then
  Law of cosines
  Let and then calculate dot product ofwith itself.
  
  
  C² A² + B² - 2AB cosθ
• Cross product of two vectors
  The cross product of two vectors is define by
  
  whereis a unit vector(vector of length 1) pointing perpendicular to the plane of and . Of course there are two directions perpendicular to any plane “in” and “out.”
  The ambiguity is resolved by the right-hand rule:
  let your fingers point in the direction of first vector and curl around (via the smaller angle) toward the second; then your thumb indicates the direction of. (In figure  points into the page;  points out of the page)
  The cross product is distributive,
  
  but not commutative.
  In fact,
  Geometrically,  is the area of the parallelogram generated by  and 
  If two vectors are parallel, their cross product is zero.
  In particularfor any vector

Vector Algebra: Component Form

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

Triple Products

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 so-called BAC-CAB 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,

Infinitesimal Displacement Vector

The infinitesimal displacement vector, from (x,y,z) to ( x + dx, y + dy, z + dz), is

Area Element

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:

Cylindrical Polar Coordinates

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,