please explain:

collinearity

proving collinearity

for the high school level

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- Aug 31st 2006, 10:55 AMligekronvectors
please explain:

collinearity

proving collinearity

for the high school level - Aug 31st 2006, 12:08 PMOReillyQuote:

Originally Posted by**ligekron**

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 PMSoroban
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 $

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

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