Question

Three points $A\left(\overline{a}\right),B\left(\overline{b}\right),C\left(\overline{c}\right)$ are collinear if and only if?

• A. $(\overline{b}-\overline{a})\times (\overline{c}-\overline{a})=0$
• B. $(\overline{b}-\overline{a})\times (\overline{c}-\overline{a})=1$
• C. $(\overline{b}-\overline{a})\cdot (\overline{c}-\overline{a})=0$
• D. $(\overline{b}-\overline{a})\cdot (\overline{c}-\overline{a})=1$

Answer 1

The correct option is A $(\overline{b}-\overline{a})\times (\overline{c}-\overline{a})=0$

Consider the given points $A\left(\overline{a}\right),B\left(\overline{b}\right),C\left(\overline{c}\right)$.
These three points determine two vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$
We know that, "two vectors $a,b$ are collinear if and only if $\overrightarrow{a}\times \overrightarrow{b}=0$".
Therefore two vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are collinear if and only if $\overrightarrow{AB}\times \overrightarrow{AC}=0$
$\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}=\overline{b}-\overline{a}$ and $\overrightarrow{AC}=\overrightarrow{OC}-\overrightarrow{OA}=\overline{c}-\overline{a}$
We have, two vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are collinear if and only if $\overrightarrow{AB}\times \overrightarrow{AC}=0$
$\Rightarrow$ two vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are collinear if and only if $(\overline{b}-\overline{a})\times (\overline{c}-\overline{a})=0$
Since the three points determine two vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$, we conclude that
Three points $A\left(\overline{a}\right),B\left(\overline{b}\right),C\left(\overline{c}\right)$ are collinear if and only if $(\overline{b}-\overline{a})\times (\overline{c}-\overline{a})=0$.