Prove the cosine rule using vectors

Yr 12 Specialist Mathematics:

Triangle ABC where (these are vectors):

AB = a

BC = b

CA = c

such that a + b = -c

Prove the cosine rule, |c|^{2}= |a|^{2} + |b|^{2} -2 |a|.|b| cosB using vectors

So far, I've been able to derive |c|^{2}= |a|^{2} + |b|^{2} **+**2 |a|.|b| cosB, with a positive not a negative. I used dot product rules where c.c = |(-a-b)^{2}|cosB. I'm a bit lost, and could really use some help on how to get the answer! :D

And this is my first time on these forums XD Could someone please tell me if there are symbols we can use? For pi, square root, things like that?

Much appreciation

-iamapineapple

Re: Prove the cosine rule using vectors

Re: Prove the cosine rule using vectors

Thanks and all, but that's the question I have to answer. "Prove it using vectors"......

Re: Prove the cosine rule using vectors

Quote:

Originally Posted by

**iamapineapple** Yr 12 Specialist Mathematics:

Triangle ABC where (these are vectors):

AB = a

BC = b

CA = c

such that a + b = -c

Prove the cosine rule, |c|^{2}= |a|^{2} + |b|^{2} -2 |a|.|b| cosB using vectors

You have several errors there.

From the given, $\displaystyle \cos(B)=\frac{-\vec{a}\cdot\vec{b}}{\|-\vec{a}\|\|\vec{b}\|}$.

$\displaystyle \|\vec{c}\|^2=(-\vec{a}-\vec{b})\cdot(-\vec{a}-\vec{b})=\|(\vec{a}\|^2+\|(\vec{b}\|^2+2\vec{a} \cdot \vec{b}$