If $(a_1 b_1 + a_2 b_2 + a_3 b_3)\lt0$ and $(a_1 c_1 + a_2 c_2 + a_3 c_3)\lt0$; what's about $(b_1 c_1 + b_2 c_2 + b_3 c_3)$?

Question

I found that problem:

Let's suppose $(a_1 b_1 + a_2 b_2 + a_3 b_3)\lt0$ and $(a_1 c_1 + a_2 c_2 + a_3 c_3)\lt0$; what's about $(b_1 c_1 + b_2 c_2 + b_3 c_3)$?

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I have no other infos about that problem because I found it on a sticky notes in an old copybook..
Only that it can't be $\lt0$ (it's only a statement, without any proof).

I'm really courious about, but I can't find any solution.

Answer 1

Let $$\overrightarrow{a}=<1,0,-1>,\overrightarrow{b}=<-1,0,\epsilon_1>,\overrightarrow{c}=<\epsilon_2,0,1>,$$ where $\epsilon_1,\epsilon_2$ are arbitrary real numbers between $0$ and $\frac 12$. Then $$\overrightarrow{a}\cdot\overrightarrow{b}=-1-\epsilon_1<0~{\rm and~}\overrightarrow{a}\cdot\overrightarrow{c}=\epsilon_2-1<0,$$ but $$\overrightarrow{b}\cdot\overrightarrow{c}=-\epsilon_2+\epsilon_1$$ can be positive, zero, or negative.

Answer 2

It can be greater than, less than, or equal to zero. I picked an example for each possibility

Greater than: $a_1=a_2=a_3=1, b_1=b_2=b_3=c_1=c_2=c_3=-1$

Less than: $a_1=a_2=a_3=b_1=c_2=1, b_2=b_3=c_1=c_3=-1$

Equal to: $a_1=a_2=1, a_3=b_2=b_3=c_1=c_3=0, b_1=c_2=-1$