Question

The three vectors $\overrightarrow{A}=3\hat{i}-2\hat{j}+\hat{k},\overrightarrow{B}=\hat{i}-3\hat{j}+5\hat{k}$ and $\overrightarrow{C}=2\hat{i}-\hat{j}+4\hat{k}$ form

• A. An equilateral triangle
• B. An isosceles triangle
• C. A right angle triangle
• D. No triangle

Answer 1

The correct option is C A right angle triangle

$\begin{array}{c}\overrightarrow{A}=3\hat{i}-2\hat{j}+\hat{k}\\ \overrightarrow{B}=\hat{i}-3\hat{j}+5\hat{k}\\ \overrightarrow{C}=2\hat{i}-\hat{j}+4\hat{k}\\ |\overrightarrow{A}|=\sqrt{{\left(3\right)}^2+{\left(-2\right)}^2+{\left(1\right)}^2}=\sqrt{14}\\ |\overrightarrow{B}|=\sqrt{{\left(1\right)}^2+{\left(-3\right)}^2+{\left(5\right)}^2}=\sqrt{35}\\ |\overrightarrow{C}|=\sqrt{{\left(2\right)}^2+{\left(-1\right)}^2+{\left(4\right)}^2}=\sqrt{21}\\ |\overrightarrow{B}{|}^2=35\\ |\overrightarrow{A}{|}^2+|\overrightarrow{C}{|}^2=14+21=35\end{array}$

Since $|\overrightarrow{B}{|}^2=|\overrightarrow{A}{|}^2+|\overrightarrow{C}{|}^2$

Therefore the given vectors form a right-angled triangle.