If $2 \hat{i}+\hat{j}-\hat{k}, \hat{i}-3 \hat{j}+5 \hat{k}$ and $-3 \hat{i}+4 \hat{j}+4 \hat{k}$ are the…

If $2 \hat{i}+\hat{j}-\hat{k}, \hat{i}-3 \hat{j}+5 \hat{k}$ and $-3 \hat{i}+4 \hat{j}+4 \hat{k}$ are the position vectors of three points A, B and C respectively, then
  1. $\mathrm{ABC}$ is a right angled triangle
  2. $\mathrm{ABC}$ is an isosceles triangle
  3. $\mathrm{A}, \mathrm{B}, \mathrm{C}$ are collinear points
  4. $\mathrm{ABC}$ is a scalene triangle

Solution

$\begin{aligned} \text { (d) } \overrightarrow{O A}=2 \hat{i}+\hat{j}-\hat{k}, \overrightarrow{O B}=\hat{i}-3 \hat{j}+5 \hat{k}, O C=-3 \hat{i}+4 \hat{j}+4 \hat{k} \\ \overrightarrow{A B}=\overrightarrow{O B}-\overrightarrow{O A}=-\hat{i}+4 \hat{j}+6 \hat{k} \Rightarrow|\overrightarrow{A B}|=\sqrt{1+16+36}=\sqrt{53} \\ \overrightarrow{B C}=\overrightarrow{O C}-\overrightarrow{O B}=-4 \hat{i}+7 \hat{j}-\hat{k} \Rightarrow|\overrightarrow{B C}|=\sqrt{16+49+1}=\sqrt{66} \\ \overrightarrow{C A}=\overrightarrow{O A}-\overrightarrow{O C}=5 \hat{i}-3 \hat{j}-5 \hat{k} \Rightarrow|\overrightarrow{C A}|=\sqrt{25+9+25}=\sqrt{59} \\ \quad|\overrightarrow{A B}| \neq|\overrightarrow{B C}| \neq|\overrightarrow{C A}| \end{aligned}$ $\therefore \triangle A B C$ is a scalene triangle.

Asked in: AP EAMCET 2023 (17 May Shift 1)

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