Question

Show that the points $A(1,2,7),B(2,6,3)$ and $C(3,10,-1)$ are collinear .

Answer 1

The given points are $A(1,2,7),B(2,6,3)$ and $C(3,10,-1)$.

$\therefore \overrightarrow{AB}=(2-1)\hat{i}+(6-2)\hat{j}+(3-7)\hat{k}=\hat{i}+4\hat{j}-4\hat{k}$

$\overrightarrow{BC}=(3-2)\hat{i}+(10-6)\hat{j}+(-1-3)\hat{k}=\hat{i}+4\hat{j}=4\hat{k}$

$\overrightarrow{AC}=(3-1)\hat{i}+(10-2)\hat{j}+(-1-7)\hat{k}=2\hat{i}+8\hat{j}-8\hat{k}$

$|\overrightarrow{AB}|=\sqrt{1^2+4^2+(-4{)}^2}=\sqrt{1+16+16}=\sqrt{33}$

$|\overrightarrow{BC}|=\sqrt{1^2+4^2+(-4{)}^2}=\sqrt{1+16+16}=\sqrt{33}$

$|\overrightarrow{AC}|=\sqrt{2^2+8^2+8^2}=\sqrt{4+64+64}=\sqrt{132}=2\sqrt{33}$

$\therefore |\overrightarrow{AC}|+|\overrightarrow{AB}|+|\overrightarrow{BC}|$

Hence, the given points $A,B$, and $C$ are collinear.