Show that the points A (3,2,-4), B (9,8,-10) and C (-2,-3,1) are … - Vidyadip

Question

Show that the points $A (3,2,-4), B (9,8,-10)$ and $C (-2,-3,1)$ are collinear.

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Answer

Let $\bar{a}, \bar{b}, \bar{c}$ be the position vectors of the points
$
\begin{aligned}
A & =(3,2,-4), B =(9,8,-10), C =(-2,-3,1) \\
\therefore \bar{a} & =3 \hat{i}+2 \hat{\jmath}-4 \hat{k} \\
\bar{b} & =9 \hat{i}+8 \hat{\jmath}-10 \hat{k} \\
\bar{c} & =-2 \hat{i}-3 \hat{\jmath}+\hat{k} \\
\overline{ AB } & =\bar{b}-\bar{a}=6 \hat{i}+6 \hat{\jmath}-6\hat{k}
\end{aligned}
$
$
\overline{ AC }=\bar{c}-\bar{a}=-5 \hat{i}-5 \hat{j}+5 \hat{k}
$
The direction ratios of the lines along $\overline{ AB }$ and $\overline{ AC }$ are proportional.
$\therefore$ Lines $A B$ and $A C$ are parallel and point $A$ is common. Therefore points A, B, C are collinear.

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