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MarkusPfundstein

Angle between two vectors - the law of cosines - derivation problem

7 posts in this topic

Hello math community,

i am studying for math at the moment and I am stuck a bit with the derivation of the law of cosines. First of all, sorry if i mention some things wrong, I study in german so it could be that we name some things different. :-)

My problem is that i don't know how to go on after a certain step. It would be cool if you could point me in the right direction .

Problem:

Derive the law of cosines from a triangle made of 2 vectors

[b]a [/b]= { ax, ay az }
[b]b = [/b]{ bx, by, bz }

[b]c[/b] = [b]b - a[/b]

All lengths are unknown and arbitrary.

So, we are looking for [b]c[/b], [b]alpha, beta & gamma[/b]

[b]alpha [/b]is between c and b
[b]beta [/b]is between c and a
[b]gamma [/b]is between a and b

the longest side is c which can be separated into [b]p [/b]and [b]q [/b]which will give me two right angled triangles.

From this information i can easily get

cos alpha = p / b -> [b]p = b * cos alpha[/b]

cos beta = q / a -> [b]q = a * cos beta[/b]

[b]c = p + q[/b]

[b]c = b * cos alpha + a * cos beta[/b]

so far so good ^-^

In the book is written that I[b] shall do the same for the sides a and b[/b]. And that is were i get stuck. All the time i am calculating the same stuff on and on again but i am not able to go one step further. It seems that i am missing some variable or s.th.

Lets say the [b]opposite leg of alpha is h[/b]


so b = sqrt ( p * p + h * h )

p = cos alpha * b

h = ??? <- how

i ??


The goal is to get

c*c = a*a + b*b - 2ab*cos [b]gamma[/b]
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I'm not sure if this is correct, so please double-check:
b = sqrt(p*p + h*h)
b^2 = p^2 + h^2
b^2 - p^2 = h^2
h = sqrt(b^2 - p^2)
h = sqrt(b^2 - (cos(alpha) * b)^2)
h = b * sqrt(1 - (cos(alpha))^2)
h = b * sqrt((sin(alpha))^2)
h = b * sin(alpha)
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Since you are working with vectors perhaps you should take a look at using the dot product, i.e. a.b = |a||b| cos(theta)
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Hey , thanks for both answers..

I had as well the solution with h = b * sin alpha but the challenge is to do it without the sin :-)

Also no dot products ...:-(

But thanks a lot for your answers!!!!
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Law of cosines states [img]http://upload.wikimedia.org/wikipedia/en/math/4/8/6/4869f4439e926aed6336ce9519f47ed7.png[/img] so I cannot see what's wrong with using [img]http://upload.wikimedia.org/wikipedia/en/math/a/5/e/a5e20c1b655833c0a260cf668c1757a8.png[/img]

Anyways, your [i][b]c = b * cos alpha + a * cos beta[/b][/i] looks like Trigonometry proof from Wikipedia (source: [url="http://en.wikipedia.org/wiki/Law_of_cosines#Using_trigonometry"]http://en.wikipedia.org/wiki/Law_of_cosines#Using_trigonometry[/url]).
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:-) yeah but sometimes its hard to filter out the right information.. i was a bit stuck in my head .. now its pretty easy of course.. i was confused because of the book..

thanks a lot though
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