Relations Involving the Directional Cosines


by Septim
Tags: cosines, directional, involving, relations
Septim
Septim is offline
#1
Feb26-12, 05:31 PM
P: 121
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
ss.JPG  
Phys.Org News Partner Science news on Phys.org
Cougars' diverse diet helped them survive the Pleistocene mass extinction
Cyber risks can cause disruption on scale of 2008 crisis, study says
Mantis shrimp stronger than airplanes
Tinyboss
Tinyboss is offline
#2
Feb27-12, 04:46 PM
P: 234
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
tiny-tim is offline
#3
Feb27-12, 05:28 PM
Sci Advisor
HW Helper
Thanks
tiny-tim's Avatar
P: 26,167
Greetings Septim!

if you've done dot-products, then:

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

Septim
Septim is offline
#4
Feb28-12, 05:42 PM
P: 121

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.
tiny-tim
tiny-tim is offline
#5
Feb29-12, 01:46 AM
Sci Advisor
HW Helper
Thanks
tiny-tim's Avatar
P: 26,167
Hi Septim!

(just got up )
Quote Quote by Septim View Post
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
Septim is offline
#6
Feb29-12, 12:37 PM
P: 121
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.


Register to reply

Related Discussions
Probability question involving recurrence relations Calculus & Beyond Homework 1
Relations bet. Groups, from Relations between Resp. Presentations. Linear & Abstract Algebra 1
Problem involving set and relations Calculus & Beyond Homework 1
Double integration of functions involving bessel functions and cosines/sines Calculus 0
Integral of exponential involving sines and cosines Calculus 1