(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Ok, so I need to differentiate the following

theta = arccos(((r_{1}-r_{2}).(r_{3}-r_{2}))/(||r_{1}-r_{2}||*||r_{3}-r_{2}||))

With regards to r_{1}, r_{2}and r_{3}

r_{1}, r_{2}and r_{3}are three dimensional vectors. "." is the scalar/dot product and * is ordinary multiplication.

2. Relevant equations

Definition of scalar product: a.b = ||a||*||b||*cos(theta)

Where theta is the angle between the vectors

3. The attempt at a solution

So I figured the chain rule be a good first attemt. The formula looks like this:

theta = f(R) = g(h(R)/k(R))

where I used simply usedRinstead of r_{1},r_{2},r_{3}.

The derivatives then look like

f ' = g'(R)*(h'*k - h*k')

I know that g' = -1/sqrt(1-(h(R)/k(R))^{2})

The problem is that I'm unsure of how do differentiate the expressions for h and k. The approach I used was as to look at them as function of ordinary variables.

Differentiate with regards to r1:

h' = r_{0}.(r_{3}-r_{2})

where r_{0}= (1,1,1) and is the differentiation of r1 with regard to itself (Is this correct?)

In the same way; k' = ||r_{3}-r_{2}||

The expressions for the derivatives with regard to r_{3}are the same, just replace r_{1}with r_{3}(and vice versa)

For the differentiation with regards to r_{2}:

h' = - r_{0}.(r_{3}-r_{2}) - r_{0}.(r_{1}-r_{2})

k' = - ||r_{3}-r_{2}|| - ||r_{1}-r_{2}||

So naturally, my question is if this is correct (and I assume it is not, for I have never done anything like this before)? If not, where did I go wrong?

Thank you for your help.

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# Homework Help: Differention of vectors in the scalar/dot product definition

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