A question about analytic geometry (vectors) Let $a , b, c$ be three vectors such that $a+3b+c = o$ where $o$ is $(0,0,0)$ vector. we also know $|a| = 3 , |b| = 4 , |c| =6$. we want to find $a.b + a.c + b.c$
I know we can solve it by saying $a+b+c = -2b$ and then squaring the sides. and at last we get the answer 3/2 . But my problem is that is it even possible for $|a| = 3 , |b| = 4 , |c| =6$ to be such that $a+3b+c = o$ ? I think it is not possible because of triangle inequality. If it is possible, how? and If it is not, why do we get an answer for it? 

As Lord Shark of the Unknown pointed out in the comments, it's not possible to have three such vectors sum up to zero. But nevertheless the calculation is possible: it is just a series of logical passages which do not yet lead to a contradiction, but surely enough if you continued to use these data in other you might end up with some absurdity.
It's like saying, "let's assume that there exists a number $x$ such that $|x|>1$ and $2x=0$." You still can deduce from the first piece of data that $|2x|>2$ or that $|x|+3>4$...  
