## Relations Involving the Directional Cosines

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.
Attached Thumbnails

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 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.
 Blog Entries: 27 Recognitions: Gold Member Homework Help Science Advisor Greetings Septim! if you've done dot-products, then: a.b = (a1i + a2j + a3k).(b1i + b2j + b3k) = … ?

## Relations Involving the Directional Cosines

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.

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