Why a timelike vector and a null vector cannot be orthogonal?

[crit]

Why a timelike vector and a null vector cannot be orthogonal?
Isn't a null vector orthogonal to any vector, by definition? Anyway, each component of a vector is multiplied by zero, so in the end the sum is zero.

 

[dextercioby]

Null 4-vector:
n^{\mu};n^{2}=:n^{\mu}n_{\mu}=0 (1)

Timelike 4-vector:
l^{\mu};l^{2}=:l^{\mu}l_{\mu}<0 (2)

Prove that
l^{\mu}n_{\mu} \neq 0(3)

HINT:Use components and the property of the 'cosine' function.


Daniel.

P.S.Esti varza...

 

[dextercioby]

HINT:'cosine' appears in the expression of the scalar product between those vectors (space components).Pay attention with the metric...

Daniel.