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Agnishom

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From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

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Vector Algebra

The question is : The sum, difference and product of 2 non-zero vectors are mutually perpendicular. Then what is the value of A.B ?

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'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

Bob

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Registered: 2010-06-20

Posts: 10,643

Re: Vector Algebra

By product, I'm assuming the vector or cross product.  This may take a little while. 

Bob

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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
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Agnishom

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From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

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Re: Vector Algebra

I'm assuming the same thing too.

One obvious result is that both these vectors are perpendicular. And A.B=0 but I am not sure.

In Mathematics, you do not understand things, you get used to them.

---

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: Vector Algebra

Is that right?  Just because a and b are perp. does that make a+b and a-b perp ?

B

I'm going to call them (ax,ay,az) etc and slog through the three cases and hope something drops out.

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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
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Agnishom

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From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

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Re: Vector Algebra

What does the B in the middle of your post mean?

Is that right?

It feels so.

---

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

Bob

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Registered: 2010-06-20

Posts: 10,643

Re: Vector Algebra

hi Agnishom,

No great mystery.  I was in a hurry so B was short for Bob.  I have an answer, but like the answer to Life, the Universe and Everything, I don't think you're going to like it. 

To save my Latexing I'm going to write the vectors as rows not columns.  And no overlines.

then if A + B is perpendicular to A - B:

So A and B must lie on the circumference of a sphere, centred on the origin.

A-B is the chord joining A to B.

The points (0,0,0), A, B and A+B would normally form a parallelogram, but as |A| = |B| these points make a rhombus.  A+B and A-B are the diagonals and so they bisect each other at right angles.

And the four points lie in the same plane.

So what about A x B ?  It will be a vector, through (0,0,0) that is perpendicular to both A and B and hence to all vectors in the plane generated by A and B as base vectors.   

Such as A+B and A-B!

So telling you that the product is also perpendicular to the other two doesn't give you any new information and any two points that are the same distance from the origin will satisfy the given properties.

So A perpendicular B would work, but it is just a special case.  It is also possible to pick two points so that A is not perpendicular to B.

So what does that means for the scalar product A.B ?

Since |A| can be any size and the cosine can take any value from -1 to + 1, that means that the scalar product can take any finite value, + or - 

Whoops!

Bob

edit: let's test that. 

Suppose you wanted the answer to be 100.

Let A = (10,10,0) and B = (0,10,10)

A + B = (10,20,10) ; A - B = (10,0,-10) and AxB = (100,-100,100)

(A + B).(A - B) = 100 - 100 = 0
(A + B).(AxB) = (10,20,10) . (100,-100,100) = 1000 - 2000 + 1000 = 0
(A - B).(AxB) = (10,0,-10) . (100,-100,100) = 1000 - 1000 = 0

So these two points have the right properties.

And finally

A.B = (10,10,0).(0,10,10) = 100

further edit:  My family asked me to do this one.

Let A = (√42,√42,0) and B = (0,√42,√42)

A + B = (√42,2√42,√42) ; A - B = (√42,0,-√42) and AxB = (42,-42,42)

(A + B).(A - B) = 42 - 42 = 0
(A + B).(AxB) = 42√42 -84√42 + 42√42 = 0
(A - B).(AxB) = 42√42 - 42√42 = 0

So these two points have the right properties.

And finally

A.B = √42.√42 = 42 

Last edited by Bob (2014-06-06 06:08:23)

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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
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

Website

Re: Vector Algebra

Hmmm

Thanks

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'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: Vector Algebra

Did you see the edits?

Bob

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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
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

Website

Re: Vector Algebra

Now I have.

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'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.