Vectors

Settings

Let

• $P$ be an arbitrary reference point.
• $\triangle ABC$ be a triangle.
• $\vec{a} = \overrightarrow{PA}$, $\vec{b} = \overrightarrow{PB}$, $\vec{c} = \overrightarrow{PC}$

Remark: $O$ is reserved for circumcenter.

Centroid

Verify that $(\vec{a} + \vec{b} + \vec{c})/3$ satisfy the constraints.

Orthocenter

Let

• $H$ be the point of intersection of two altitudes $AA_H$ and $BB_H$
• $\vec{h} = \overrightarrow{PH}$

Add these two equations together.

Preface

There’s a simpler derivation using

1. the geometric defintion
2. linearity from the definition
3. orthogonality of the canonical basic vectors $\vec{i}$, $\vec{j}$ and $\vec{k}$.

If I had read that, I wouldn’t have type this document in LaTeX. The following content was transcribed from a PDF file that I made three days ago.

Content

Recall: $\sum_{i=m}^{n} a_{i}=a_{m}+a_{m+1}+a_{m+2}+\cdots +a_{n-1}+a_{n},$ where $m$, $n$ are integers.

Example: $\sum_{i=3}^6 i^2 = 3^2 + 4^2 + 5^2 + 6^2 = 86$