Question

if a, b, c are mutually perpendicular unit vectors then find the value of [a b c]²

Answer 1

Consider a,b,c as complex numbers which can be represented in the cartesian plane similar to vectors.

Let a=cosx + i sinx

b=cosy + i siny

c=cosz + i sinz

all the above complex numbers have modulus 1.

since a+b+c=0

hence sigma(cosx) + i sigma(sinx)=0

hence real and imaginary both parts are separately zero.

a^2+b^2+c^2=cos2x + cos2y + cos2z+ i(sin2x + sin2y + sin2z)=0

expand (a+b+c)^2=0 to get the result