<s>[INST] 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? 
 [/INST] Dot the equation with each of the vectors
\begin{eqnarray*}
\mid a \mid^2 +3 a \cdot b +a \cdot c =0 \\
a \cdot b +3 \mid b \mid^2 +b \cdot c =0 \\
 a \cdot c +3 b \cdot c+\mid c \mid^2  =0 \\
\end{eqnarray*}
Now add the first and third equations and subtract the second $2(a \cdot b +a \cdot c +b \cdot c)+9+36-48=0$ ...
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