seven-dimensional vectors and its … · goyal s k, new pattern iit-jee (2013), arihant prakshan...

NJACSRR/JUL-DEC2014/VOL-1/ISS-1/A3 ISSN:2394 - 4870

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NATIONAL JOURNAL OF ARTS, COMMERCE & SCIENTIFIC

RESEARCH REVIEW

SEVEN-DIMENSIONAL VECTORS AND ITS PROPERTIES

HITESH KANSAL*1

PRINCE GARG*2

MONIKA YADAV*3

CSE department Student III SEM.

Dronacharya College of Engineering, Gurgaon

[email protected]

[email protected]

_____________________________________________________

ABSTRACT:-

This research paper deals with vectors in 3-dimentional space and 7-dimentional space mainly and different components for different directions, forces, velocities and various other quantities are vectors. This makes the algebra and calculus of these vector functions, the natural instrument for the engineer and physicist in solid mechanics, fluid dynamics, heat transfer and thermo dynamics, electrostatics, etc. This paper includes the way of finding products of two or more vectors in both scalar and vector form. In three dimensions (as opposed to higher dimensions), geometrical ideas become,

influential, enrich the theory, and many geometrical quantities (tangents and normal) may be given by vectors.

KEY WORDS:-

Euclidean spaces, bilinear operation, anti-commutative, parallelotope&Hodgedual



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INTRODUCTION:-

A vector is a quantity that is determined by both its magnitude and its direction. A vector (arrow) has a tail, called initial point, and a tip is called its terminal point. The length (or magnitude) of a vector (length of the arrow) is also called the norm (or

Euclidean norm) of and is denoted by . In geometry, physics and engineering applications we use two kinds of quantities, i.e., scalars and vectors. A scalar is a quantity that is determined by its magnitude. In mathematics, the seven-dimensional cross product is a bilinearoperation on vectors in sevendimensionalEuclideanspaces. It assigns to any two vectors a, b in ℝ7 a vector a × b also in ℝ7. Like the cross-product in three dimensions the seven-dimensional product is anti-commutative and a × b is orthogonal to both a andb.

Geometrical application of dot product

(i) Projection of

= ab.cosα

= ( . )/a

= . â

(ii) Projection of in the direction perpendicular to

= (iii) Mechanical application

Work done =

Geometrical application of cross product

(i) Area of parallelogram = absinα =

(ii) Area of triangle =1/2 (iii) Area of triangle =

1/2 (iv) Area of abcd any quadrilateral is cross product of its

diagonals =1/2 (v) Trigonometric application (i) sine law

+ =

absin

SinC/c =sinB/b Hence SinA/a =SinB/b = SinC/c

(ii) sin(A+B) = SinACosB +CosASinB

Mechanicalapplication

(i) moment of force M =

Scalar triple product

, , are three non-zero vectors then Scalar triple product is denoted by

[ ] = . ( )

Properties

(i) [ ] = . ( )

=( ) .

(ii) = ai + bj + ck

= di + ej + fk

= li + mj + nk

. ( ) = (ai + bj + ck)

=

Similarly,

(iii) [0 ] =0

(iv) [ ] = 0

(v) [ x y ] = xy[ ]

(vi) [ + ] = [ ] + [ ]



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(1) If , , arethree non-zero vectors

Then , , will be coplanar

Iff [ ] = 0

(2) Volume of tetrahedral whose combines edges are , , is 1/6[ ]

(3) Volume of parallelepiped whose coplanar edges are , , is

V= [ ]

(4) The non- coplanar vectors , , in order form a

right handed system of vectors if [ ] <0

Vectors , , in order form a left handed system of vectors if [ ] <0

(5) [ ][ ] =

(6) [ ][ ] =

(7) [ + + + ] = 2[ ]

(8) [ ] = [ ][ ] (9) [I j k] = 1 (10) ai + bj + ck

di +ej + fk

Where , ,

[ ] = [ ]

=

) = [a ].

=

Vector triple product

1.

2.

The vector in the plane of & & perpendicular to can be suppose on

( )

Product of 4 vectors:-

Scalar product of 4 vectors

( ) =

( )

Vector product of 4 vectors

=

Reciprocal system of vectors

Let , , be three non-coplanar vectors then the vector defined as follows

)/[ ]

( )/[ ]

( )/[ ]

Are called reciprocal system w.r.t. the triad of vectors < >

Some applications

Similarly,



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Similarly all will be zero

0

[ [ ]

=

=

[ = 1

Seven-dimensional cross product

In mathematics, the seven-dimensional cross product is a bilinearoperation on vectors in sevendimensionalEuclideanspaces. It assigns to any two vectors a, b in ℝ7 a vector a × b also in ℝ7. Like the crossproduct in three dimensions the seven-dimensional product is anticommutative and a × b is orthogonal to both a andb.

Multiplication table

× e1 e2 e3 e4 e5 e6 e7

e1 0 e3 −e2 e5 −e4 −e7 e6

e2 −e3 0 e1 e6 e7 −e4 −e5

e3 e2 −e1 0 e7 −e6 e5 −e4

e4 −e5 −e6 −e7 0 e1 e2 e3

e5 e4 −e7 e6 −e1 0 −e3 e2

e6 e7 e4 −e5 −e2 e3 0 −e1

e7 −e6 e5 e4 −e3 −e2 e1 0

123, 145, 176, 257, 347, 365

The product can be given by a multiplication table, such as the one above. This table, due to Caley, gives the product of basis vectors ei and ej for each i, j from 1 to 7. For example from the table

The table can be used to calculate the product of any two vectors. For example to calculate the e1 component of x × y the basis vectors that multiply to produce e1 can be picked out to give

Definition

The cross product on a EuclideanspaceV is a bilinearmap from V × V to V, mapping vectors x and y in V to another vector x × y also in V, where x × y has the properties

orthogonality:

,

magnitude:

1. Anticommutativity:



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,

2. Scalartripleproduct:

3. Malcevidentity:

Coordinate expressions

To define a particular cross product, an orthonormal basis {ej} may be selected and a multiplication table provided that determines all the products {ei × ej}. One possible multiplication table is described in the Examplesection, but it is not unique. Unlike three dimensions, there are many tables because every pair of unit vectors is perpendicular to five other unit vectors, allowing many choices for each cross product.

Once we have established a multiplication table, it is then applied to general vectors x and y by expressing x and y in terms of the basis and expanding x × y through bilinearity.

× e1 e2 e3 e4 e5 e6 e7 e1 0 e4 e7 −e2 e6 −e5 −e3 e2 −e4 0 e5 e1 −e3 e7 −e6 e3 −e7 −e5 0 e6 e2 −e4 e1 e4 e2 −e1 −e6 0 e7 e3 −e5 e5 −e6 e3 −e2 −e7 0 e1 e4 e6 e5 −e7 e4 −e3 −e1 0 e2 e7 e3 e6 −e1 e5 −e4 −e2 0

124, 137, 156, 235, 267, 346, 457

Generalizations

Non-trivial binary cross products exist only in three and seven dimensions. But if the restriction that the product is binary is lifted, so products of more than two vectors are allowed, then more products are possible. As in two dimensions the product must be vector valued, linear, and anti-commutative in any two of the vectors in the product.

The product should satisfy orthogonality, so it is orthogonal to all its members. This means no more than n − 1 vectors can be used in n dimensions. The magnitude of the product should equal the volume of the parallelotope with the vectors as edges, which is can be calculated using the Gramdeterminant. So the conditions are

orthogonality:

the Gram determinant:



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The Gramdeterminant is the squared volume of the parallelotope with a1, ...,ak as edges. If there are just two vectors x and y it simplifies to the condition for the binary cross product given above, that is

With these conditions a non-trivial cross product only exists:

as a binary product in three and seven dimensions as a product of n − 1 vectors in n> 3 dimensions as a product of three vectors in eight dimensions

Conclusions:-

In this research paper you have learned about the basic of the vectors and also the various ways in which we can implement the various properties of vectors in the physics and also our daily life. In this we have studied also about a totally new type of vectors i.e. Reciprocal system of vectors this system is used to solve various types of tough and lengthy problems related to vectors can be solved in seconds. Also in this we have come to know about the Seven Dimensional Vectors, their cross product and also the various components of that in all seven dimensions

after cross product with the help of a table and the negativity or positivity of the sign is determined by a diagram shown in this paper this helps a lot to determine sign otherwise is a tough task to know the positive or negative sign.

REFERENCES:-

Goyal S K, New Pattern IIT-JEE (2013), Arihant Prakshan Publications

Agarwal Amit M,Mathematics For IIT-JEE (2013), Arihant Prakshan Publications

Vector 3-D Calculus(2013),Cengage Learning India Pvt. LtdPublications

A Das Gupta, Problems Plus In IIT Mathematics (2011),Bharti Bhavan

Pundir S. K., Sharma J. N., Khanna ML, IITMathematics For JEE (2013),Jai Prakash Nath & Co.

Jacobson Nathan, Basic Algebra I (2009),Dover Publications Inc.

Hall and Knight, Higher Algebra (2013),Arihant prakshan publications

seven-dimensional vectors and its … · goyal s k, new pattern iit-jee (2013), arihant prakshan...

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