I've got a section in my textbook about non-parallel vectors, it says:
For two non-parallel vectors a and b, if $\lambda a + \mu b = \alpha a + \beta b$
then $\lambda = \alpha $ and $\mu = \beta $
Okay I get that you can equate coefficients and solve for mu and lambda, but how are the two sides of the equation equal in the first place? How can you just equate two different vectors to each other like that? I'm just confused and i'm not entirely sure what about. I've tried googling but not much turns up.. I'd love it if someone could explain in basic terms what this equation is telling me.. (that non parallel vectors are equal?) as I've only just been introduced to this topic recently.. Thank you.
Best Answer
If you are in $\mathbb{R}^n$ of in any vector space then you can always compare two vectors. They are just two elements of a set. Equality perfectly make sense. Perhaps the following description can help you.
If you rewrite this equation, you will get $$(\lambda -\alpha)a=(\beta -\mu)b$$ which is same as $$a=(\beta -\mu)/(\lambda -\alpha)b.$$ That is a is a scalar multiple of b. Therefore if they are not parallel (if x=cy for two vectors x and y and scalar c then x and y are parallel) then the denominator should be 0 hence you get the result.