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Dot Product in Three Dimensions

Geometrically, we know that two vectors are perpendicular if the Pythagorean Theorem holds, i.e. the square of the length of (a,b,c) plus the square of the length of (x,y,z) equals the square of the length of (a,b,c) - (x,y,z) = (a-x,b-y,c-z). This means that a2 + b2 + c2 + x2 + y2 + z2 = (a-x)2 + (b-y)2 + (c-z)2 = a2 - 2ax + x2 + b2 - 2by + y2 + c2 - 2cz + z2. From this it follows that 0 = -2ax - 2by - 2cz, so ax + by + cz = 0. Thus two vectors in R3 are perpendicular if and only if their dot product is zero.
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