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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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