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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 Vector Algebra | Mathematical Methods - Physics for any general vector and its magnitude byVector Algebra | Mathematical Methods - Physics 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. MinusVector Algebra | Mathematical Methods - Physics(-Vector Algebra | Mathematical Methods - Physics)is a vector with the same magnitude as Vector Algebra | Mathematical Methods - Physics but of opposite direction.

Vector Algebra | Mathematical Methods - Physics


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

  • Addition of two vectors
    Place the tail of Vector Algebra | Mathematical Methods - Physics at the head of Vector Algebra | Mathematical Methods - Physics; the sum,Vector Algebra | Mathematical Methods - Physics+Vector Algebra | Mathematical Methods - Physics is the vector from the tail of Vector Algebra | Mathematical Methods - Physics to the head of Vector Algebra | Mathematical Methods - Physics
    Addition is commutative:  Vector Algebra | Mathematical Methods - Physics
    Addition is associative:Vector Algebra | Mathematical Methods - Physics
    To subtract a vector, add its opposite:Vector Algebra | Mathematical Methods - PhysicsVector Algebra | Mathematical Methods - Physics
  • 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:
    Vector Algebra | Mathematical Methods - Physics
    Vector Algebra | Mathematical Methods - Physics
  • Dot product of two vectors
    The dot product of two vectors is define by
    Vector Algebra | Mathematical Methods - Physics
    Vector Algebra | Mathematical Methods - Physics
    where θ is the angle they form when placed tail to tail. Note that Vector Algebra | Mathematical Methods - Physics. is itself a scalar.
    The dot product is commutative,
    Vector Algebra | Mathematical Methods - Physics
    and distributive,Vector Algebra | Mathematical Methods - Physics
    Geometrically Vector Algebra | Mathematical Methods - Physics is the product of A times the projection of Vector Algebra | Mathematical Methods - Physics along Vector Algebra | Mathematical Methods - Physics (or the product of B times the projection of Vector Algebra | Mathematical Methods - Physics)
    If the two vectors are parallel,Vector Algebra | Mathematical Methods - Physics
    If two vectors are perpendicular, then Vector Algebra | Mathematical Methods - Physics
    Law of cosines
    Let Vector Algebra | Mathematical Methods - Physicsand then calculate dot product ofVector Algebra | Mathematical Methods - Physicswith itself.
    Vector Algebra | Mathematical Methods - Physics
    Vector Algebra | Mathematical Methods - Physics
    C2 A2 + B2 - 2AB cosθ
  • Cross product of two vectors
    Vector Algebra | Mathematical Methods - PhysicsThe cross product of two vectors is define by
    Vector Algebra | Mathematical Methods - Physics
    whereVector Algebra | Mathematical Methods - Physicsis a unit vector(vector of length 1) pointing perpendicular to the plane of Vector Algebra | Mathematical Methods - Physicsand Vector Algebra | Mathematical Methods - Physics. 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 ofVector Algebra | Mathematical Methods - Physics. (In figure Vector Algebra | Mathematical Methods - Physics points into the page; Vector Algebra | Mathematical Methods - Physics points out of the page)
    The cross product is distributive,
    Vector Algebra | Mathematical Methods - Physics
    but not commutative.
    In fact, Vector Algebra | Mathematical Methods - Physics
    Geometrically, Vector Algebra | Mathematical Methods - Physics is the area of the parallelogram generated by Vector Algebra | Mathematical Methods - Physics andVector Algebra | Mathematical Methods - Physics 
    If two vectors are parallel, their cross product is zero.
    In particularVector Algebra | Mathematical Methods - Physicsfor any vector Vector Algebra | Mathematical Methods - Physics

Vector Algebra: Component Form

Let Vector Algebra | Mathematical Methods - PhysicsandVector Algebra | Mathematical Methods - Physics be unit vectors parallel to the x, y and z axis, respectively. An arbitrary

vectorVector Algebra | Mathematical Methods - Physicscan be expanded in terms of these basis vectors
Vector Algebra | Mathematical Methods - Physics
Vector Algebra | Mathematical Methods - Physics
The numbers Ax , Ay , and Az are called component of Vector Algebra | Mathematical Methods - Physics geometrically, they are the projections ofVector Algebra | Mathematical Methods - Physicsalong the three coordinate axes.
(i) Rule: To add vectors, add like components.
Vector Algebra | Mathematical Methods - Physics
(ii) Rule: To multiply by a scalar, multiply each component.
Vector Algebra | Mathematical Methods - Physics
Because Vector Algebra | Mathematical Methods - Physics and Vector Algebra | Mathematical Methods - Physics are mutually perpendicular unit vectors
Vector Algebra | Mathematical Methods - Physics
Accordingly, Vector Algebra | Mathematical Methods - Physics
(iii) Rule: To calculate the dot product, multiply like components, and add.
In particular, Vector Algebra | Mathematical Methods - Physics
Similarly, Vector Algebra | Mathematical Methods - Physics
Vector Algebra | Mathematical Methods - Physics
(iv) Rule: To calculate the cross product, form the determinant whose first row is Vector Algebra | Mathematical Methods - Physicswhose second row is Vector Algebra | Mathematical Methods - Physics (in component form), and whose third row isVector Algebra | Mathematical Methods - Physics 
Vector Algebra | Mathematical Methods - Physics

Example 1: Find the angle between the face diagonals of a cube.

The face diagonals Vector Algebra | Mathematical Methods - Physics and Vector Algebra | Mathematical Methods - Physics are
Vector Algebra | Mathematical Methods - Physics
Vector Algebra | Mathematical Methods - Physics
Vector Algebra | Mathematical Methods - Physics
Vector Algebra | Mathematical Methods - Physics


Example 2: Find the angle between the body diagonals of a cube.

The body diagonals Vector Algebra | Mathematical Methods - Physics and Vector Algebra | Mathematical Methods - Physics are
Vector Algebra | Mathematical Methods - Physics
Vector Algebra | Mathematical Methods - Physics
Vector Algebra | Mathematical Methods - Physics
Vector Algebra | Mathematical Methods - Physics


Example 3: Find the components of the unit vector nˆ perpendicular to the plane shown in the figure.

The vectors Vector Algebra | Mathematical Methods - Physics and Vector Algebra | Mathematical Methods - Physics can be defined as
Vector Algebra | Mathematical Methods - Physics
Vector Algebra | Mathematical Methods - Physics


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:Vector Algebra | Mathematical Methods - Physics
Geometrically Vector Algebra | Mathematical Methods - Physicsis the volume of the parallelepiped generated by Vector Algebra | Mathematical Methods - PhysicsandVector Algebra | Mathematical Methods - Physics , sinceVector Algebra | Mathematical Methods - Physicsis the area of the base, and Vector Algebra | Mathematical Methods - Physics is the altitude. Evidently,
Vector Algebra | Mathematical Methods - Physics
Vector Algebra | Mathematical Methods - Physics
In component form Vector Algebra | Mathematical Methods - Physics
Note that the dot and cross can be interchanged: Vector Algebra | Mathematical Methods - Physics
(ii) Vector triple product: Vector Algebra | Mathematical Methods - Physics
The vector triple product can be simplified by the so-called BAC-CAB rule:
Vector Algebra | Mathematical Methods - Physics
Position, Separation and Displacement Vectors
Vector Algebra | Mathematical Methods - Physics
Vector Algebra | Mathematical Methods - Physics
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:
Vector Algebra | Mathematical Methods - Physics
Its magnitude, Vector Algebra | Mathematical Methods - Physics is the distance from the origin,
andVector Algebra | Mathematical Methods - Physics is a unit vector pointing radially outward.
Note: In electrodynamics one frequently encounters problems involving two points-

typically, a source point, Vector Algebra | Mathematical Methods - Physics where an electric charge is located, and a field point,Vector Algebra | Mathematical Methods - Physics, at which we are calculating the electric or magnetic field. We can define separation vector

from the source point to the field point by Vector Algebra | Mathematical Methods - Physics
Vector Algebra | Mathematical Methods - Physics
Its magnitude is Vector Algebra | Mathematical Methods - Physics
and a unit vector in the direction from Vector Algebra | Mathematical Methods - Physics
In Cartesian coordinates, Vector Algebra | Mathematical Methods - Physics
Vector Algebra | Mathematical Methods - Physics
Vector Algebra | Mathematical Methods - Physics


Infinitesimal Displacement Vector Vector Algebra | Mathematical Methods - Physics

The infinitesimal displacement vector, from (x,y,z) to ( x + dx, y + dy, z + dz), is
Vector Algebra | Mathematical Methods - Physics

Area ElementVector Algebra | Mathematical Methods - Physics

For closed surface area element is perpendicular to the surface pointing outwards as shown in figure below.

Vector Algebra | Mathematical Methods - Physics

(i) For x = 2 plane,Vector Algebra | Mathematical Methods - Physics
(ii) For x = 0 plane,Vector Algebra | Mathematical Methods - Physics
(iii) For y = 2 plane,Vector Algebra | Mathematical Methods - Physics
(iv) For y = 0 plane,Vector Algebra | Mathematical Methods - Physics
(v) For z = 2 plane,Vector Algebra | Mathematical Methods - Physics
(vi) For z = 0 plane,Vector Algebra | Mathematical Methods - Physics
For open surface area element is shown in figure below (use right hand rule)
Vector Algebra | Mathematical Methods - Physics
Volume ElementVector Algebra | Mathematical Methods - Physics

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.

Vector Algebra | Mathematical Methods - Physics

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
andVector Algebra | Mathematical Methods - Physics
The range of r is 0 → ∞, θ goes from 0 → π , and ∅ goes from0 → 2π .
Figure shows three unit vectorsVector Algebra | Mathematical Methods - Physics , pointing in the direction of increase of the

corresponding coordinates. They constitute an orthogonal (mutually perpendicular) basis

set (just likeVector Algebra | Mathematical Methods - Physics), and any vector Vector Algebra | Mathematical Methods - Physics can be expressed in terms of them in the usual

way:
Vector Algebra | Mathematical Methods - Physics
Ar , Aθ , and A∅ are the radial, polar and azimuthal components of Vector Algebra | Mathematical Methods - Physics .
Infinitesimal Displacement Vector Vector Algebra | Mathematical Methods - Physics

An infinitesimal displacement in the Vector Algebra | Mathematical Methods - Physics direction is simply dr (figure a), just as an

infinitesimal element of length in the x direction is dx :
dlr = dr
On the other hand, an infinitesimal element of length in theVector Algebra | Mathematical Methods - Physics direction (figure b) is r dθ
dlθ = rdθ
Similarly, an infinitesimal element of length in the Vector Algebra | Mathematical Methods - Physics direction (figure c) is rsin θd∅
dl = r sinθd∅
Thus, the general infinitesimal displacement dl is
Vector Algebra | Mathematical Methods - Physics
This plays the role (in line integrals, for example) that =Vector Algebra | Mathematical Methods - Physics played in

Cartesian coordinates.
Vector Algebra | Mathematical Methods - Physics
Area ElementVector Algebra | Mathematical Methods - Physics
Vector Algebra | Mathematical Methods - Physics
If we are integrating over the surface of a sphere, for instance,
then r is constant, whereas θ and ∅ change, so
Vector Algebra | Mathematical Methods - Physics
on the other hand, if the surface lies in the xy plane, then θ is
constant (θ = π/2) while r and ∅ vary, then
Vector Algebra | Mathematical Methods - Physics
Volume Element(d/τ)
The infinitesimal volume element dτ, in spherical coordinates, is the product of the three infinitesimal displacements:
Vector Algebra | Mathematical Methods - Physics
Transformation of Vector to Spherical Polar
We can transform any vector in Cartesian coordinates to Spherical polar coordinate as Vector Algebra | Mathematical Methods - Physics
Thus
Vector Algebra | Mathematical Methods - Physics
where x = rsin θcos∅, y = rsinθsin∅, z = r cosθ
and use table given below:
Vector Algebra | Mathematical Methods - Physics

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
Vector Algebra | Mathematical Methods - Physics
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
Vector Algebra | Mathematical Methods - Physics
soVector Algebra | Mathematical Methods - Physics
and volume element is dτ = r dr dФ, dz.
We can transform any vectorVector Algebra | Mathematical Methods - Physics in Cartesian coordinates to cylindrical coordinates asVector Algebra | Mathematical Methods - Physics
Thus,
Vector Algebra | Mathematical Methods - Physics

The document Vector Algebra | Mathematical Methods - Physics is a part of the Physics Course Mathematical Methods.
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FAQs on Vector Algebra - Mathematical Methods - Physics

1. What is vector algebra?
Ans. Vector algebra is a branch of mathematics that deals with the manipulation and analysis of vectors. It involves operations such as addition, subtraction, scalar multiplication, dot product, cross product, and finding the magnitude and direction of vectors.
2. How is vector algebra used in IIT JAM?
Ans. Vector algebra is an essential concept in IIT JAM as it is used to solve problems related to physics, mathematics, and other subjects. Many questions in the exam require the application of vector algebra to solve problems involving forces, motion, electromagnetic fields, and geometrical concepts.
3. What are the basic operations in vector algebra?
Ans. The basic operations in vector algebra include addition, subtraction, scalar multiplication, dot product, and cross product. Addition and subtraction involve combining or subtracting the corresponding components of two vectors. Scalar multiplication scales a vector by a scalar value. The dot product gives a scalar value representing the projection of one vector onto another, while the cross product gives a vector perpendicular to both input vectors.
4. How do I find the magnitude of a vector using vector algebra?
Ans. To find the magnitude of a vector using vector algebra, you need to take the square root of the sum of the squares of its components. For a vector A = (A1, A2, A3), the magnitude |A| is given by |A| = sqrt(A1^2 + A2^2 + A3^2).
5. Can vector algebra be used in three-dimensional space?
Ans. Yes, vector algebra can be used in three-dimensional space. Vectors in three-dimensional space have three components, representing their projections along the x, y, and z axes. The operations of vector algebra, such as addition, subtraction, and cross product, are extended to three dimensions by considering the corresponding components of the vectors.
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