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**Vector Planes & Orthogonality -- Help!**

I must be doing something really stupid, and I'll kick myself when you point it out, but I'm having difficulty with this question:

**Find the unit normal to the plane x + 2y – 2z = 15. What is the distance of the plane from the origin?**

OK, so I know I need three points in the plane to identify it. I found points:

A = (15,0,0)

B = (15,1,1)

C = (13,2,1)

They all seem to satisfy the equation, right?

So now I find two nonparallel vectors in this plane:

A->B = 0

**i**+ 1

**j**+ 1

**k**

A->C = -2

**i**+ 2

**j**+ 1

**k**

They do indeed join those points, right?

So now I want to find an orthogonal vector, so I take the cross product by calculating the determinant of the 3x3 matrix:

[

**i**

**j**

**k**]

[0 1 1]

[-2 2 1]

= (1-2)

**i**- (-2-0)

**j**+ (0+2)

**k**

= -

**i**+ 2

**j**+ 2

**k**

This vector ought to be orthogonal to vectors A->B, A->C and B->C (and their negatives). But I try to test it with the usual method of adding up the parts and seeing if they result in zero, using A->C as my comparison vector:

(-2 * -1) + (2 * 2) + (1 * 2) = 8

Why doesn't this equal zero? I have a feeling that the middle (

**j**) term is wrong, because if that was (2 * -2) then the whole thing would equal zero, but I don't see where I've got my signs wrong in the method.

Any help would be much appreciated.

Cheers!

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