Definition:Between (Geometry)/N-dimensional Euclidean space Intuition
Jump to navigation
Jump to search
This page has been identified as a candidate for refactoring of medium complexity. Until this has been finished, please leave {{Refactor}} in the code.
New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Refactor}} from the code. |
It has been suggested that this page be renamed. To discuss this page in more detail, feel free to use 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:
- $\size {\map \cos {\angle {BAC} } } = 1$
Hence, from Cosine Formula for Dot Product, we should have:
- $\size {\vec{AB} \cdot \vec{AC} } = \norm {\vec {AB} } * \norm {\vec {AC} }$
This article, or a section of it, needs explaining. In particular: What does $*$ mean? Convolution? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code. |
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 $AC$.