How to find the quadrant of a dot product

Question

Consider the vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}, \mathbf{d}$ in the picture below:

enter image description here

Consider the list of dot products: dot(a,b) , dot(a,c) , dot(a,d) , dot(b,c) , dot(b,d) , dot(c,d) (latex didn't work for this)

For each dot product, enter in positive if it's positive, negative if it's negative, and $0$ if it's equal to zero. Answer with the resulting list.

I tried
0 negative negative positive negative 0

0 positive negative positive negative 0

0 positive negative positive positive negative

but they r all wrong.

Answer 1

Hint: The dot product of two vectors $a,\, b\in\mathbb R^n$ is the same as $$\langle a,\,b\rangle=\lVert a\rVert \lVert b\rVert \cos\varphi$$ where $\varphi$ is the angle enclosed by $a$ and $b$. Now, for which angles $\varphi$ is $\cos\varphi$ negative, for which positive?

Answer 2

More specifically, for $\vec{a}\cdot\vec{b}>0\implies \cos\theta>0,$ then it is QI or QIV. For $\vec{a}\cdot\vec{b}<0\implies \cos\theta<0,$ then it is QII or QIII.