Using vectors to prove the cosine rule...?

Question

I have this question that states "if the triangle ABC is representated be vectors such that AB=a, BC=b and CA=c, where a+b=-c
prove the cosine |c|^2= |a|^2 + |b|^2 - 2|a||b|*cosC "

How do I do this? I'm not very good at the whole proving things with vectors lol, so thanks for any help :)

Answer 1

Hello there,



Could you please ensure that the question states a + b = -c? This does not seem to make sense because if you draw a triangle labelled as ABC, you can see that vectors AB + BC ≠ -AC.



Personally, I would work with a - b = c because if you draw these vectors and add them, you can see that AB + (-BC) = CA. If you need help with this, I will give you a hint by saying that B is "between" points A and C. Point A should be the most southern point and C the most northern.



Apply dot product to (a - b = c) to prove the cosine law:



(a - b) • (a - b)= c • c



|a|^2 - 2|a||b|cosC - |b|^2 = |c|^2



There you go! Observe that the |a|^2 and |b|^2 do not have cosines attached because the angle between parallel vectors is 0° and cos(0) = 1.



Hope this helps!

Answer 2

Cosine Rule Proof

Answer 3

since AB=a, BC=b and CA=c,



substitute the variables...



CA^2=AB^2+BC^2-2(AB)(CA)



So you have



CA= the square root of AB^2+BC^2-2(AB)(CA)



So you just have to type AB^2+BC^2-2(AB)(CA) into a calculator or solve it on paper... there will be numbers for each AC BC and AC.



that is how you solve for the law of cosines. (SSS)

Answer 4

It is U•V, not U x V Let W = U – V ||W||² = ||U – V||² = (U – V)•(U – V) = ||U||² + ||V||² – 2U•V The Law of cosines states that ||W||² = ||U||² + ||V||² – 2||U||||V||cosΘ Comparing the last two equations, we see that 2U•V = 2||U||||V||cosΘ, or cosΘ = U•V/(||U||||V||)

Answer 5

should it equal zero?







uhh I shouldn't even get two points!