# vectors

• Aug 31st 2006, 10:55 AM
ligekron
vectors

collinearity

proving collinearity

for the high school level
• Aug 31st 2006, 12:08 PM
OReilly
Quote:

Originally Posted by ligekron

collinearity

proving collinearity

for the high school level

Collinearity is refering to position of points in line.

Given points are collinear if line contains all of them. If line doesn't contain all of them then they are non-collinear.
• Aug 31st 2006, 01:05 PM
Soroban
Hello, ligekron!

Quote:

Please explain: (1) Collinearity, (2) Proving collinearity
for the high school level

Since your heading says "Vectors", I assume a vector approach is in order.

Three points $\displaystyle A,\,B,\,C$ are collinear (lie on a straight line)
if any two of their vectors $\displaystyle \{\overrightarrow{AB},\;\overrightarrow{BC},\; \overrightarrow{AC}\}$ are scalar multiples of each other.

$\displaystyle \text{Example: }\;A(2,3),\;B(3,5),\;C(5,9)$

$\displaystyle \text{Let }\vec{u} \:=\:\overrightarrow{AB} \:= \:\langle 3,5\rangle - \langle 2,3\rangle \:=\:\langle 1,2\rangle$

$\displaystyle \text{Let }\vec{v}\:=\:\overrightarrow{BC} \:=\:\langle 5,9\rangle - \langle 3,5\rangle \:=\:\langle 2,4\rangle$

$\displaystyle \text{Since }\vec{v} = 2\vec{u}\text{, points }A,\;B,\,C\text{ are collinear.}$