$(a^2 + b^2 + c^2)(m^2 + n^2 + k^2) - (am + bn + ck)^2 = (an - bm)^2 + (ak - cm)^2 + (bk - cn)^2$
$a^2m^2 + a^2n^2 + a^2k^2 + b^2m^2 + b^2n^2 + b^2k^2 + c^2m^2 + c^2n^2 + c^2k^2 = a^2m^2 + a^2n^2 + a^2k^2 + b^2m^2 + b^2n^2 + b^2k^2 + c^2m^2 + c^2n^2 + c^2k^2 - (a^2m^2 + b^2n^2 + c^2k^2 + 2ambn + 2bnck + 2amck) = a^2m^2 + a^2n^2 + a^2k^2 + b^2m^2 + b^2n^2 + b^2k^2 + c^2m^2 + c^2n^2 + c^2k^2 - a^2m^2 - b^2n^2 - c^2k^2 - 2ambn - 2bnck - 2amck = (a^2m^2 - a^2m^2) + (b^2n^2 - b^2n^2) + (c^2k^2 - c^2k^2) + a^2n^2 + a^2k^2 + b^2m^2 + b^2k^2 + c^2m^2 + c^2n^2 - 2ambn - 2bnck - 2amck = (a^2n^2 - 2ambn + b^2m^2) + (a^2k^2 - 2bnck + c^2m^2) + (b^2k^2 - 2bnck + c^2n^2) = (an - bm)^2 + (ak - cm)^2 + (bk - cn)^2$
$(an - bm)^2 + (ak - cm)^2 + (bk - cn)^2 = (an - bm)^2 + (ak - cm)^2 + (bk - cn)^2$