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) = (ax,by,cz). This means that a^{2} + b^{2} + c^{2} + x^{2} + y^{2} + z^{2} = (ax)^{2} + (by)^{2} + (cz)^{2} = a^{2}  2ax + x^{2} + b^{2}  2by + y^{2} + c^{2}  2cz + z^{2}. From this it follows that0 = 2ax  2by  2cz , soax + by + cz = 0 . Thus two vectors in R^{3} are perpendicular if and only if their dot product is zero.

