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:

$\size {\map \cos {\angle {BAC} } } = 1$

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

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

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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 $AC$.