Question

verify whether the points (3,-2,4),(1,0,-2)and (-1,2,-8) are collinear ​

Answer 1

$\textbf{Given:}$

$\textsf{Points are (3,-2,4), (1,0,-2) and (-1,2,-8)}$

$\textbf{To check:}$

$\textsf{whether the given points are collinear or not}$

$\textbf{Solution:}$

$\textsf{First we find, equation of line joining (3,-2,4) and (1,0,-2)}$

$\mathsf{\dfrac{x-x_1}{x_2-x_1}=\dfrac{y-y_1}{y_2-y_1}=\dfrac{z-z_1}{z_2-z_1}}$

$\mathsf{\dfrac{x-3}{1-3}=\dfrac{y+2}{0+2}=\dfrac{z-4}{-2-4}}$

$\mathsf{\dfrac{x-3}{-2}=\dfrac{y+2}{2}=\dfrac{z-4}{-6}}$

$\mathsf{Put\;(x,y,z)=(-1,2,-8)}$

$\mathsf{\dfrac{-1-3}{-2}=\dfrac{2+2}{2}=\dfrac{-8-4}{-6}}$

$\mathsf{\dfrac{-4}{-2}=\dfrac{4}{2}=\dfrac{-12}{-6}}$

$\mathsf{2=2=2}$

$\textsf{The equation is satisfied}$

$\therefore\textsf{The third point lies on the line joininig of the first two points}$

$\textsf{Hence, the given 3 points are collinear}$

Answer 2

$\textbf{Given:}$

$\textsf{Points are (3,-2,4), (1,0,-2) and (-1,2,-8)}$

$\textbf{To check:}$

$\textsf{whether the given points are collinear or not}$

$\textbf{Solution:}$

$\textsf{First we find, equation of line joining (3,-2,4) and (1,0,-2)}$

$\mathsf{\dfrac{x-x_1}{x_2-x_1}=\dfrac{y-y_1}{y_2-y_1}=\dfrac{z-z_1}{z_2-z_1}}$

$\mathsf{\dfrac{x-3}{1-3}=\dfrac{y+2}{0+2}=\dfrac{z-4}{-2-4}}$

$\mathsf{\dfrac{x-3}{-2}=\dfrac{y+2}{2}=\dfrac{z-4}{-6}}$

$\mathsf{Put\;(x,y,z)=(-1,2,-8)}$

$\mathsf{\dfrac{-1-3}{-2}=\dfrac{2+2}{2}=\dfrac{-8-4}{-6}}$

$\mathsf{\dfrac{-4}{-2}=\dfrac{4}{2}=\dfrac{-12}{-6}}$

$\mathsf{2=2=2}$

$\textsf{The equation is satisfied}$

$\therefore\textsf{The third point lies on the line joininig of the first two points}$

$\textsf{Hence, the given 3 points are collinear}$