Triple Product of three vectors

There are two types of triple product of vectors:

(a)   Scalar triple product.

(b)   Vector triple product.

Scalar Triple Product

Definition :

              let u , v, and w  be three vectors.

The scalar triple product of vectors u , v, and w  is defined by u . ( v ×w) or v . ( w ×u)

or w . (u×v).

The scalar triple product u.(v×w) is written as   u. ( v×w) = [u v w ].

Note :

 (1) The value of the scalar triple product depends upon the cycle  order of the vectors , but is independent of the position of the dot and cross. So the dot and cross , may be interchanged without altering the value  i.e;

               u . ( v ×w )= u× ( v .w ) =  [u v w]

  (2) the value of the product changes if the order  is non cyclic i.e;

   u . ( v ×w)= - u.( w×v ) = -[u v w]

Analytical Expression of u .( v ×w):                                

               

         

which is called the determinant formula of scalar triple product of u , v and w in component form.

where u = a₁i + b₁j + c₁k , v = a₂i + b₂j + c₂k and w = a₃i + b₃j + c₃k

THE VOLUME OF THE PARALLELEPIPED

Statement :

The scalar triple product u× ( v .w) represents  the volume of the parallelepiped having u , v and w as its conterminous edges.

Proof: As ,

  u× ( v .w) =  u× v       w  cosӨ

where u ×v    = area of the parallelogram with two adjacent sides u and v.

 and    w  cosӨ  = height of the parallelepiped.

Therefore,   u× ( v .w) =  u × v     w  cosӨ    =( area of parallelepiped)( height)

                                   = volume of parallelepiped.

Similarly by taking the base planeformed by v and w , we have

The volume of parallelepiped  =  (v×  w) . u

And by taking the base plane formed by w and u ,we have

The volume of parallelepiped = (   w ×  u)  .v

Thus ( u  × v  )  .w  =( v  ×w)  .u  =  (w ×  u )  .v   =  V

"CONCURRENCY OF THREE LINES"

 

“Concurrency of Three Lines”

When two or more straight lines meet in a point,they are said to be “Concurrent” and their meeting point is called the “Point of Concurrency”.

     Where d is the point of concurrency.

Let the equations of the three intersecting lines l₁, l₂ and l₃ be respectively.

(i)     l₁: a₁x+b₁y+c₁=0

(ii)    l₂:a₂x+b₂y+c₂=0

(iii)   l₃: a₃x+b₃y+c₃=0

If the above lines are concurrent then condition of concurrency for lines is written in the determinant form as:                                

                                                      

      

      
The above condition is not sufficient to ensure that the three given lines are concurrent .However, it can be shown that, if the above determinant vanishes, then either the given lines are parallel or concurrent.

Here one has to investigate the coefficients; if coefficients of the variables are not same then lines are concurrent.

problem

Q:  Show that the following lines are concurrent. Also find their point of concurrency.

(i) x-y = 6, 4y +22 = 3x  and  6x+5y+8= 0

solution:

                                        i.         l₁ : x-y = 6…………..eq(i)

                                      ii.         l₂ : 4y+22 = 3x………eq(ii)

                                    iii.         l₃ : 6x+5y+8 = 0…….eq(iii)

 Therefore equation (i) will be;

ð x= 6+y……..(A)

Putting in eq (ii); we get;

ð 4y+22 = 3(6+y)

ð 4y+22 =18+3y

ð 4y-3y= 18-22

ð y = -4

Putting in eq (A); we get;

ð x= 6+(-4)

ð x=2

Therefore   (2,-4) are the coordinates of point of intersection of l₁ and l₂. If line 

l₃: 6x+5y+8=0 also passes through the same  point then:

ð 6(2)+5(-4)+8 = 0

ð 12-20+8=0

ð -8+8=0---------àwhich is true.

Therefore it is proved that lines are concurrent and (2,-4) is the point of concurrency.

CURRENT DEVELOPMENTS IN MATHEMATICS

THE 13"TH CONFERENCE ON  CURRENT DEVELOPMENT IN MATHEMATICS IS LIKELY TO HELD ON NOVEMBER 18---19,2011( FRIDAY AND SATUARDAY) AT SCIENCE CENTRE,LECTURE HALL TBA, HARVARD UNIVERSITY , CAMBRIDGE, MA , 02138.

THE CONFEENCE IS ORGANISEDBY DAVID JERSION , RICHARD STANLEY, THOMAS MROWALA(MIT) , HORNG- TZER YAU, MARK KISIN., SHING-TUNG YAN(HARVARD).

FREE REGISTRATION IS OPEN .LIMITED FUNDING TO HELP DEFRAY TRAVEL EXPENSES IS AVAILABLE FOR GRADUATE STUDENTS AND RECENT PhD'S .
A GRADUATE STUDENT WILL HAVE TO SEND BRIEF LETTER OF RECOMMENDATION FROM A FACULTY MEMBER TO EXPLAIN THE RELEVANCE OF THE CONFERENCE TO STUDIES OR RESEARCH.

THERE WILL BE A BANQUET ON FRIDAY NOVEMBER 18,2011, 7 PM IN THE ROYAL EAST RESTAURANT 791 MAIN STREET, CAMBRIDGE MA 02139. BANQUET TICKETS WILL BE DISTRIBUTED AT THE CONFERENCE ON FRIDAY.

THE EVENT SPONSORS ARE THE NATIONAL SCIENCE FOUNDATION, HARVARD UNIVERSITY AND MIT.

THE CDM 2011 SPEAKERS ARE  JEREMY KAHN( STONY BROOK UNIVERSITY), RICHARD KENYON ( BROWN UNIVERSITY ) ROBIN PENMANTLE( UNIVERSITY OF PENNSILVANIA ), JEREMY QUASTED ( UNIVERSITY OF TORONTO)AND ASHKAY VENKATESH( STANDFORD UNIVERSITY).

FOR ANY QUERY REGARDING THE EVENT , DO WRITE TO MAUREEN ARMSTRONG AT        maureen@math.harvard.edu.

OPERATION RESEARCH ANALYST

HOW TO BECOME AN OPERATION RESEARCH ANALYST?

 

INTRODUCTION TO THE FIELD:
Operations research analyst (O.R.), also known as management science are usually employed in the industry.
People in this field help organization coordinate and operate in the most efficient manner by applying math skills to organizational problems.

THE CHALLENGE:
When employers are seeking people, they are looking for two primary skills:
First, they are looking for someone who can think logically and work well with people.
Second, they are looking for people who are able to use a computer and do some programming.

The later skill is very important because computers are used extensively for quantitative analysis. Finally, of course, they are looking for math majors too!

Standard O.R. techniques such as LINEAR PROGRAMMING, CRITICAL PATH ANALYSIS,SIMULATION, STATISTICAL DECISION THEORY, QUEUING THEORYAND INVENTORY CONTROL THEORY,MATHEMATICAL MODELING  may be used.

Specialist work may be in the production, marketing or financial planning areas. 

However, Strong background in mathematics and computer should be enough to get you a ticket into that field.

GETTING THE JOB:

If you are interested, you would normally get a bachelor in O.R. or if you don't have a bachelor in O.R., but you have one in mathematics, you would try to get a master in O.R.

To the best of my knowledge, I don't believe certification is a requirement at this moment in order to become an operations research analyst.

Over time, after you have gained enough experience, you may become an independent consultant or some other high level managerial position

what's the use of Mathematics?

The most common questions that high school students raise during class : when am I ever going to use math in the real world?

The idea behind this post is to motivate students to pursue mathematics-based careers, by helping them understand just how much math actually influences and shapes society. check it out following video to have more clear vision of Mathematics.