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 $A(1,2,7),B(2,6,3),and \space C(3,10,-1).$
$∴\overrightarrow{AB}=(2-1)\hat{i}+(6-2)\hat{j}+(3-7)\hat{k}=\hat{i}+4\hat{j}-4\hat{k}$
$\overrightarrow{BC}=(3-2)\hat{i}+(10-6)\hat{j}+(-1-3)\hat{k}=\hat{i}+4\hat{j}-4\hat{k}$
$\overrightarrow{AC}=(3-1)\hat{i}+(10-2)\hat{j}+(-1-7)\hat{k}=2\hat{i}+8\hat{j}-8\hat{k}$
$|\overrightarrow{AB}|=\sqrt{1^{2}+4^{2}+(-4)^{2}}=\sqrt{1+16+16}=\sqrt{33}$
$|\overrightarrow{BC}|=\sqrt{1^{2}+4^{2}+(-4)^{2}}=\sqrt{1+16+16}=\sqrt{33}$
$|\overrightarrow{AC}|=\sqrt{2^{2}+8^{2}+8^{2}}=\sqrt{4+64+64}=\sqrt{132}=2\sqrt{33}$
$∴|\overrightarrow{AB}|=|\overrightarrow{AB}|+|\overrightarrow{BC}|$
Hence,the given points $A,B,and \space C$ are collinear.