From 2183c3a2db899f3f99c23c0f1bb8c92de0b00b4b Mon Sep 17 00:00:00 2001
From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Fri, 19 Oct 2018 11:16:15 +1100
Subject: [PATCH] geometric proofs with vectors

---
 spec/vectors.md | 13 +++++++++++++
 1 file changed, 13 insertions(+)

diff --git a/spec/vectors.md b/spec/vectors.md
index a2122b0..8fa7208 100644
--- a/spec/vectors.md
+++ b/spec/vectors.md
@@ -108,5 +108,18 @@ Vector resolute of $\boldsymbol{a}$ in direction of $\boldsymbol{b}$ is magnitud
 
 $$\boldsymbol{u}={{\boldsymbol{a}\cdot\boldsymbol{b}}\over |\boldsymbol{b}|^2}\boldsymbol{b}=\left({\boldsymbol{a}\cdot{\boldsymbol{b} \over |\boldsymbol{b}|}}\right)\left({\boldsymbol{b} \over |\boldsymbol{b}|}\right)=(\boldsymbol{a} \cdot \hat{\boldsymbol{b}})\hat{\boldsymbol{b}}$$
 
+## Vector proofs
+
+**Concurrent lines -** $\ge$ 3 lines intersect at a single point  
+**Collinear points -** $\ge$ 3 points lie on the same line
+
+Useful vector properties:
+
+- If $\boldsymbol{a}$ and $\boldsymbol{b}$ are parallel, then $\boldsymbol{b}=k\boldsymbol{a}$ for some $k \in \mathbb{R} \setminus \{0\}$
+- If $\boldsymbol{a}$ and $\boldsymbol{b}$ are parallel with at least one point in common, then they lie on the same straight line
+- Two vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ are perpendicular if $\boldsymbol{a} \cdot \boldsymbol{b}=0$
+- $\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2$
+
+
 
 
-- 
2.43.2