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

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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 $AC$.