Relations Involving the Directional Cosines

[Septim]

Greetings,

I wonder if a proof of the relation between the directional cosines of two vectors and cosine between two vectors is available? In order to clarify what I meant I put a screen shot from Vector and Tensor Analysis by Hay.

 

[Tinyboss]

Verify in the easy case where one of your vectors is (a,0,0) for some a>0. Since every other case can be gotten from the easy one by a rotation (which preserves the angle between the vectors), and since orthogonal matrices preserve the expression involving the direction cosines (use the fact that their rows are unit-length vectors), you're done.

 

[tiny-tim]

Greetings Septim!

if you've done dot-products, then:

a.b = (a₁i + a₂j + a₃k).(b₁i + b₂j + b₃k) = … ?

 

[Septim]

Thanks for the replies. Tinyboss I will try the method you suggested but I am a bit unfamiliar with matrices. Tiny-tim the author derives uses the expression you see on the attachment to convert the abstract form of the dot product into the component form. My question is that how can he relate cosines with directional cosines. I am still open for other suggestions.

 

[tiny-tim]

Hi Septim!

(just got up :zzz:)

… the author derives uses the expression you see on the attachment to convert the abstract form of the dot product into the component form. My question is that how can he relate cosines with directional cosines.

I'm not following you.

Which equations are you referring to? ​

 

[Septim]

After the expression of the dot product in abstract form, that is Eq.(7.1); the author expresses the cosine between the two vectors in terms of the direction cosines of the individual vectors. This equation is indented; however it does not have an equation number. I actually wonder how that equation can be derived. Forgive me for the late reply by the way.