Physics  >  Mathematical Models  >  Vector Algebra

Vector Algebra - Notes | Study Mathematical Models - Physics

1 Crore+ students have signed up on EduRev. Have you?

Vector Operations

Vector quantities have both direction as well as magnitude such as velocity, acceleration,

force and momentum etc. We will use Vector Algebra - Notes | Study Mathematical Models - Physics for any general vector and its magnitude byVector Algebra - Notes | Study Mathematical Models - 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 - Notes | Study Mathematical Models - Physics(-Vector Algebra - Notes | Study Mathematical Models - Physics)is a vector with the same magnitude as Vector Algebra - Notes | Study Mathematical Models - Physics but of opposite direction.

Vector Algebra - Notes | Study Mathematical Models - Physics


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

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

Vector Algebra: Component Form

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

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

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

The face diagonals Vector Algebra - Notes | Study Mathematical Models - Physics and Vector Algebra - Notes | Study Mathematical Models - Physics are
Vector Algebra - Notes | Study Mathematical Models - Physics
Vector Algebra - Notes | Study Mathematical Models - Physics
Vector Algebra - Notes | Study Mathematical Models - Physics
Vector Algebra - Notes | Study Mathematical Models - Physics


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

The body diagonals Vector Algebra - Notes | Study Mathematical Models - Physics and Vector Algebra - Notes | Study Mathematical Models - Physics are
Vector Algebra - Notes | Study Mathematical Models - Physics
Vector Algebra - Notes | Study Mathematical Models - Physics
Vector Algebra - Notes | Study Mathematical Models - Physics
Vector Algebra - Notes | Study Mathematical Models - Physics


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

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

typically, a source point, Vector Algebra - Notes | Study Mathematical Models - Physics where an electric charge is located, and a field point,Vector Algebra - Notes | Study Mathematical Models - 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 - Notes | Study Mathematical Models - Physics
Vector Algebra - Notes | Study Mathematical Models - Physics
Its magnitude is Vector Algebra - Notes | Study Mathematical Models - Physics
and a unit vector in the direction from Vector Algebra - Notes | Study Mathematical Models - Physics
In Cartesian coordinates, Vector Algebra - Notes | Study Mathematical Models - Physics
Vector Algebra - Notes | Study Mathematical Models - Physics
Vector Algebra - Notes | Study Mathematical Models - Physics


Infinitesimal Displacement Vector Vector Algebra - Notes | Study Mathematical Models - Physics

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

Area ElementVector Algebra - Notes | Study Mathematical Models - Physics

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

Vector Algebra - Notes | Study Mathematical Models - Physics

(i) For x = 2 plane,Vector Algebra - Notes | Study Mathematical Models - Physics
(ii) For x = 0 plane,Vector Algebra - Notes | Study Mathematical Models - Physics
(iii) For y = 2 plane,Vector Algebra - Notes | Study Mathematical Models - Physics
(iv) For y = 0 plane,Vector Algebra - Notes | Study Mathematical Models - Physics
(v) For z = 2 plane,Vector Algebra - Notes | Study Mathematical Models - Physics
(vi) For z = 0 plane,Vector Algebra - Notes | Study Mathematical Models - Physics
For open surface area element is shown in figure below (use right hand rule)
Vector Algebra - Notes | Study Mathematical Models - Physics
Volume ElementVector Algebra - Notes | Study Mathematical Models - 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 - Notes | Study Mathematical Models - 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 - Notes | Study Mathematical Models - Physics
The range of r is 0 → ∞, θ goes from 0 → π , and ∅ goes from0 → 2π .
Figure shows three unit vectorsVector Algebra - Notes | Study Mathematical Models - Physics , pointing in the direction of increase of the

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

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

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

An infinitesimal displacement in the Vector Algebra - Notes | Study Mathematical Models - 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 - Notes | Study Mathematical Models - Physics direction (figure b) is r dθ
dlθ = rdθ
Similarly, an infinitesimal element of length in the Vector Algebra - Notes | Study Mathematical Models - Physics direction (figure c) is rsin θd∅
dl = r sinθd∅
Thus, the general infinitesimal displacement dl is
Vector Algebra - Notes | Study Mathematical Models - Physics
This plays the role (in line integrals, for example) that =Vector Algebra - Notes | Study Mathematical Models - Physics played in

Cartesian coordinates.
Vector Algebra - Notes | Study Mathematical Models - Physics
Area ElementVector Algebra - Notes | Study Mathematical Models - Physics
Vector Algebra - Notes | Study Mathematical Models - Physics
If we are integrating over the surface of a sphere, for instance,
then r is constant, whereas θ and ∅ change, so
Vector Algebra - Notes | Study Mathematical Models - 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 - Notes | Study Mathematical Models - Physics
Volume Element(d/τ)
The infinitesimal volume element dτ, in spherical coordinates, is the product of the three infinitesimal displacements:
Vector Algebra - Notes | Study Mathematical Models - Physics
Transformation of Vector to Spherical Polar
We can transform any vector in Cartesian coordinates to Spherical polar coordinate as Vector Algebra - Notes | Study Mathematical Models - Physics
Thus
Vector Algebra - Notes | Study Mathematical Models - Physics
where x = rsin θcos∅, y = rsinθsin∅, z = r cosθ
and use table given below:
Vector Algebra - Notes | Study Mathematical Models - 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 - Notes | Study Mathematical Models - 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 - Notes | Study Mathematical Models - Physics
soVector Algebra - Notes | Study Mathematical Models - Physics
and volume element is dτ = r dr dФ, dz.
We can transform any vectorVector Algebra - Notes | Study Mathematical Models - Physics in Cartesian coordinates to cylindrical coordinates asVector Algebra - Notes | Study Mathematical Models - Physics
Thus,
Vector Algebra - Notes | Study Mathematical Models - Physics

The document Vector Algebra - Notes | Study Mathematical Models - Physics is a part of the Physics Course Mathematical Models.
All you need of Physics at this link: Physics
18 docs|24 tests
Use Code STAYHOME200 and get INR 200 additional OFF
Use Coupon Code
18 docs|24 tests
Download as PDF

Download free EduRev App

Track your progress, build streaks, highlight & save important lessons and more!

Related Searches

Vector Algebra - Notes | Study Mathematical Models - Physics

,

Exam

,

video lectures

,

practice quizzes

,

Vector Algebra - Notes | Study Mathematical Models - Physics

,

Viva Questions

,

pdf

,

Previous Year Questions with Solutions

,

ppt

,

shortcuts and tricks

,

study material

,

MCQs

,

Extra Questions

,

Semester Notes

,

past year papers

,

Free

,

Important questions

,

Objective type Questions

,

mock tests for examination

,

Vector Algebra - Notes | Study Mathematical Models - Physics

,

Sample Paper

,

Summary

;