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#1 2016-10-28 04:54:20

Zeeshan 01
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Registered: 2016-07-22
Posts: 648

Vector help 1

Hi I have vectors topic problems
My 1 st question is prove that vectorially cos (a+b)=cosacosb-sinasinb
A and b are alpha and beta respectively plese help and also tell by diagram every single step
Thx....


MZk

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#2 2016-10-29 01:11:14

bob bundy
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Registered: 2010-06-20
Posts: 8,168

Re: Vector help 1

hi Zeeshan 01

This proof first establishes a matrix for a rotation of angle a, and then uses matrix multiplication of vectors to derive the formula.

KlQC5RQ.gif

This diagram shows a circle of unit radius.  Some points are shown.  OA makes an angle of a with the x axis.  OB makes an angle of b.

Because these points are on a circle of radius 1 =>  OA = OB = 1 and these lines are the relative hypotenuses so making the coordinates of the points

A = (cosa , sina) and B = (cosb, sinb)  I have not written cos(a) to avoid the double brackets.

Consider the points (1,0) and (0,1).  These can be written together as vectors like this:

Consider the rotation R of 'a' degrees that maps (1,0) onto A and (0,1) onto C. [and also B onto D] By comparing the coordinates C = (-sina , cosa)

So the matrix for R is given by [R times start vectors = finish vectors]

Now apply R to the vector for B to get the coordinates of D (where D makes an angle of a+b with the x axis).

Thus we have both compound angle formulas in one go:

cos(a+b) = cosa.cosb - sina.sinb

and

sin(a+b) = sina.cosb + cosa.sinb

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#3 2016-10-29 04:58:33

Zeeshan 01
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Registered: 2016-07-22
Posts: 648

Re: Vector help 1

Can this proof become more simple??? WHERE this circle comes from


MZk

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#4 2016-10-29 20:08:28

bob bundy
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Re: Vector help 1

hi Zeeshan 01

I've seen the formula proved using Euclidean geometry.  It is much more complicated.  The proof above is actually just one line:


but to understand it you need to understand (1) how to multiply a vector by a matrix; (2) how to form a rotation matrix; (3) that points on the unit circle have simple coordinates based on the angle a line makes with the x axis.

I added all three to my proof because I wasn't sure if you know those things.  That's what makes the proof seem a long one.  I notice from your other question that you haven't yet met (4) the scalar (dot) product and (5) the vector (cross) product.  So that's five new things you need to learn if you are going to master vectors.  I'm happy to explain all but it will take many posts.

Let's start with the unit circle.

BxKYBdY.gif

angle AOE = a  OA = 1

Using sin = opp/hyp and cos = adj/hyp, we find that the coordinates of A are (cosa , sina)

This is useful in many ways.  It allows us to  extend the definition of sin and cos beyond 0-90.  It allows us to examine rotations around the origin.  For your problem it allows us to write the coordinates of B and C in a similar way.

Please post back to say this part is clearer now, or with a further question about what I have said so far.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#5 2016-10-30 01:53:15

Zeeshan 01
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Registered: 2016-07-22
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Re: Vector help 1

sin = opp/hyp what is opp i konw only base altitude hypotenus


MZk

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#6 2016-10-30 02:40:51

Mathegocart
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Registered: 2012-04-29
Posts: 1,884

Re: Vector help 1

Zeeshan 01 wrote:

sin = opp/hyp what is opp i konw only base altitude hypotenus

Opposite is the side opposite the angle specified,  adjacent is the side that is consecutive with the angle, and you know what hyp is.


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#7 2016-10-30 13:32:04

Zeeshan 01
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Registered: 2016-07-22
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Re: Vector help 1

Yes


MZk

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#8 2016-10-30 21:05:04

thickhead
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Registered: 2016-04-16
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Re: Vector help 1

Last edited by thickhead (2016-11-05 02:05:58)


{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

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#9 2016-10-31 01:57:54

Zeeshan 01
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Registered: 2016-07-22
Posts: 648

Re: Vector help 1

That's I undersand but how cos and sin come in circle that true that it make angle alpha n beta but in angle a with x axis clock wise how this come icosa+jsina??


MZk

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#10 2016-10-31 01:59:14

Zeeshan 01
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Registered: 2016-07-22
Posts: 648

Re: Vector help 1

I need circle explaination


MZk

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#11 2016-10-31 02:30:55

thickhead
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Registered: 2016-04-16
Posts: 1,086

Re: Vector help 1

Yes Zeeshan, I made a mistake. I shall correct it now.


{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

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#12 2016-10-31 02:48:45

Zeeshan 01
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Registered: 2016-07-22
Posts: 648

Re: Vector help 1

Ok


MZk

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#13 2016-11-04 20:39:16

bob bundy
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Registered: 2010-06-20
Posts: 8,168

Re: Vector help 1

hi Zeeshan 01

Why a circle?  Well why not?  It helped me make my proof.  thickhead has shown a method that doesn't use the circle, so you can take your pick.  There's no point in getting locked into this one thing.

Do you have any other questions on this topic?

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#14 2016-11-05 01:56:30

Zeeshan 01
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Registered: 2016-07-22
Posts: 648

Re: Vector help 1

I don't understand thick head method in his third line how come icosa+jsina..... how cos and sin come


MZk

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#15 2016-11-05 08:36:05

bob bundy
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Registered: 2010-06-20
Posts: 8,168

Re: Vector help 1

0zlGJ0U.gif

OP is one unit in length.  Let Q be the point on the x axis so that angle POQ = A and angle PQO = 90.

Then using trigonometry

and similarly

So the vector OP is given by

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#16 2016-11-05 16:17:00

Zeeshan 01
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Registered: 2016-07-22
Posts: 648

Re: Vector help 1

Thx this method helps me but last step how op=cosai+sin. J     
From origin to p how you add oq+pq


MZk

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#17 2016-11-05 20:12:15

bob bundy
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Registered: 2010-06-20
Posts: 8,168

Re: Vector help 1

From origin to p how you add oq+pq

This is how vector addition works.  This page will be helpful

http://www.mathsisfun.com/algebra/vectors.html

There are also links to the pages on the scalar and vector products.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#18 2016-11-05 20:22:52

Zeeshan 01
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Registered: 2016-07-22
Posts: 648

Re: Vector help 1

This is vector addition you do in op???


MZk

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#19 2016-11-05 21:33:34

bob bundy
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Registered: 2010-06-20
Posts: 8,168

Re: Vector help 1

This is vector addition you do in op???

Yes!!!  Your topic is about vectors.  I have overlined the vectors to show that they are vectors.  This is easier for me in Latex than making them bold.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#20 2016-11-06 00:05:51

Zeeshan 01
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Registered: 2016-07-22
Posts: 648

Re: Vector help 1

Ok


MZk

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