Show that two nonzero vectors V1 and V2 in R³ are orthogonal if and only if their direction cosines satisfy Cos(alpha1)Cos(alpha2)+Cos(Beta1)Cos(Beta2)+Cos(Gamma1)Cos(Gamma2) = 0

Solution;

If there are two vectors V1 and V2 as given by

V1=<a1 , b1 , c1>

V2=<a2 , b2 , c2>

Then their direction Cosines that are angles made by these two Vectors with Coordinate axis are given by

For V1=<a1,b1,c1>

Cos(alpha1) = a1/||V1||

Cos(Beta1) = b1/||V1||

Cos(Gamma1) = c1/||V1||

For V2=<a2,b2,c3>

Cos(alpha2) = a2/||V2||

Cos(Beta2) = b2/||V2||

Cos(Gamma2) = c2/||V2||

As we have to prove that vectors V1 and V2 are orthogonal by proving that their direction cosines satisfy Cos(alpha1)Cos(alpha2)+Cos(Beta1)Cos(Beta2)+Cos(Gamma1)Cos(Gamma2)=0

As we know that when two vectors are orthogonal then their dot product is zero such as

V1.V2 = 0

As

 V1.V2 = a1a2 + b1b2 + c1c2

As both vectors are orthogonal so

a1a2 + b1b2 + c1c2 = 0 ...(1)

As above we discuss direction cosines of both vectors so by using these directions Cosines we have

a1 = ||V1|| Cos(alpha1)

b1 = ||V1|| Cos(Beta1)

c1 = ||V1|| Cos(Gamma1)

And

a2 = ||V2|| Cos(alpha2)

b2 = ||V2|| Cos(Beta2)

c2 = ||V2|| Cos(Gamma2)

By putting values in equestion (1)

||V1||||V2|| Cos(alpha1) Cos(alpha2) + ||V1||||V2|| Cos(Beta1) Cos(Beta2) + ||V1||||V2||Cos(Gamma1) Cos(Gamma2) = 0

After simplification

||V1||||V2||[Cos(alpha1)Cos(alpha2)+Cos(Beta1)Cos(Beta2)+Cos(Gamma1)Cos(Gamma2)] = 0

We have

Cos(alpha1)Cos(alpha2)+Cos(Beta1)Cos(Beta2)+Cos(Gamma1)Cos(Gamma2)=0

Hence it is proved that V1 and V2 are orthogonal.