# Definition:Between (Geometry)/N-dimensional Euclidean space Intuition

Jump to navigation Jump to search
 It has been suggested that this page or section be renamed. One may discuss this suggestion on the talk page.

The intuition behind this definition comes from the fact that when we think that $B$ is between $A$ and $C$, we think of three things.

The first thing is that $A$, $B$ and $C$ are collinear.

So, $\size{\cos{(\angle{BAC})}} = 1$.

Hence, from Cosine Formula for Dot Product, we should have:

$\size {\vec{AB} \cdot \vec{AC} } = \norm{\vec{AB}} * \norm{\vec{AC}}$

Secondly, the vectors $\vec{AB}$ and $\vec{AC}$ have the same direction.

Therefore, their dot product should be positive.

So,

${\vec{AB} \cdot \vec{AC} } = \norm{\vec{AB}} * \norm{\vec{AC}}$

Thirdly, the length of $AB$ should be less than the length of $AC$.