Answer to Question #12458 in Analytic Geometry for john.george.milnor

Question

Prove identity: [a,b]^2+(a,b)^2=|a|^2 * |b|^2, where a and b are vectors

Answer 1

|[a,b]|=|a|*|b|*sin(a^b)
(a,b)=|a|*|b|*cos(a^b)
Using cos^2(x)+sin^2(x)=1
and squaring two previuos equalities we obtain
necessary result.