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Hi Randy,
Thanks for the explanation...
Just to clarify, what you did there was:
- Take cross product of 2 vectors and then took the dor product with
any one of them to show it was 0, so the angle between the must be 90
deg.
Cheers,
P
Randy Poe wrote:
> pankaj.daga@xxxxxxxxx wrote:
> > Hi everyone,
> >
> > This is a pretty high school question...
> >
> > I picked up a book on linear algebra and was going through the chapter
> > on vectors and vector multiplication...
> >
> > The dot product is very clear.
> >
> > Howeever, with the vector cross product, it says that the cross product
> > vector is always normal to the plane containing the two vectors that
> > are being multiplied.
> >
> > Though I do not question the validity of the statement. However, is
> > there a proof that shos that the cross product is always normal. I am
> > probably having a bit of a tough time visualizing the cross product in
> > the geometrical sense.
>
> The algebraic proof from the algebraic definition is simple enough,
> just symbol manipulation.
>
> Consider the dot product of (x1,y1,z1) with the cross product
> (y1*z2 - y2*z1, z1*x2 - z2*x1, x1*y2 - y1*x2)
>
> This is equal to (x1*y1*z2 - x1*y2*z1) + (y1*z1*x2 - y1*z2*x1)+
> (z1*x1*y2 - z1*y1*x2)
>
> The 1st term cancels the 4th term, the 2nd with the 5th, the
> 3rd with the 6th.
>
> So the dot product is zero. The two vectors are normal.
>
> That doesn't help your intuition much though. I learned the geometric
> definition first, that the cross-product of v1 and v2 was defined as a
> vector normal to both, direction given by right-hand rule, and
> that its magnitude was |v1| |v2| sin(theta) where theta is the angle
> between v1 and v2. It was then proven that the algebraic definition
> had these properties.
>
> - Randy
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