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This is a very noob question, but can someone please give me an example of finding the normal vector to a surface (if this is the word in English) which is defined by three points in it. I know that there are two ways - the first to find the equation of the surface and to take the coefficients, and the second - to find two vectors from the points and then to multiply them vectorly (I'm almost sure that this isn't the name for this operation x in English), but I still have a little difficulties and an example with concrete points will be very helpful!

Thank you very much in advance and please excuse me for my stupidity and bad English!

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Suppose you have points $A(0,0,0),B(0,1,0)$ and $C(1,0,0)$. We need to find the normal to the unique surface(which is a plane for 3 points) passing through these points.

Now, $\vec{AB}, \vec{AC}$ both lies in the plane and cross-product of any two non-zero vectors is a vector normal to both the vectors, and hence normal to the surface containing the vectors.

Thus required vector is $\vec{AB}$ x $\vec{AC}=(\vec j) $ x $(\vec i)=-\vec{k}$

So any scalar multiple of this vector=$\lambda \vec k$ is normal to the surface.

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So (0, 0, k) is the normal vecktor? Thank you very much! – Pretty_lemon Jan 18 '13 at 7:29

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