# Two Points and Surface Normal

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Hi, Straight up, Assignment Question. Using the equation of a 3D plane is ax + by + cz = d, where a, b, c and d are constants, what is the plane normal vector? I am to demonstrate how I can work out the normal vector. Now if you are going to response, please don't give the answer but hints. I may be way off but has it something to do with if I have 2 vectors (not parallel) lying on a plane, I can use the cross product of the vectors to determine the normal. But if I only have two points on the plane then surely that only makes one vector. But the lecture has hinted that you only need to points and using the equation of a 3D plane can determine the surface normal.

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Hint: All the information you need is right there in the equation ax + by + cz = d. The normal can be computed (or even just extracted, depending) directly from this equation; no extra vectors or cross products are needed.

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I realise that the normal is <a,b,c> of the plane equation but this is something I read but not understand.

A hint that was given was:

what would be the result of the following dot product?
n . (p1 – p2)

Use the x, y and z components of each of the points and carry out the dot product above. Can you see a pattern?

I also read that the dot product of a normal to a vector lying on a surface is zero. It is like I have the answers but do not understanding the workings.

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Also, I am aware of dot product of two vector is equal to the magitude of a times b times cos angle .... a.b = |a||b| cos o

Has that anything to do with it?

Am I anywhere close.

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You shouldn't need trigonometry for this.

a point p(x, y, z) is on the plane(a, b, c, d) if p . n = d.

from this and a * x + b * y + c * z = d, you can work out n.x, n.y, n.z.

EDIT : the plane parameters should actually be (a, b, c, -d), since a plane equation is usually in the form ax + by + cz + d = 0. But you can ignore that bit of trivia, unless you get bonus points :)

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