Question

Show that the following set of point are collinear ?
$(2,3,-4),(-1,0,5),(3,4,-7)$

Answer 1

Let the given points are

$A(2,3,-4),B(-1,0,5),C(3,4,-7)$

Then,

$\begin{array}{c}\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}=-3\hat{i}-3\hat{j}+9\hat{k}\\ \\ \overrightarrow{BC}=\overrightarrow{OC}-\overrightarrow{OB}=4\hat{i}+4\hat{j}-12\hat{k}\\ \\ \overrightarrow{AC}=\overrightarrow{OC}-\overrightarrow{OA}=\hat{i}+\hat{j}-3\hat{k}\\ \\ \left|\overrightarrow{AB}\right|=\sqrt{{\left(-3\right)}^2+{\left(-3\right)}^2+{\left(9\right)}^2}=3\sqrt{11}\\ \\ \left|\overrightarrow{BC}\right|=\sqrt{{\left(4\right)}^2+{\left(4\right)}^2+{\left(-12\right)}^2}=4\sqrt{11}\\ \\ \left|\overrightarrow{AC}\right|=\sqrt{{\left(1\right)}^2+{\left(1\right)}^2+{\left(-3\right)}^2}=\sqrt{11}\end{array}$

Since,

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

Therefore, given points are collinear.