Question

The vectors $3\hat{i}-2\hat{j}+\hat{k}$,$\hat{i}-3\hat{j}+5\hat{k}$ and $2\hat{i}+\hat{j}-4\hat{k}$ form the sides of a triangle. This triangle is?

• A. an equilateral triangle
• B. an isosceles triangle
• C. a right angled triangle
• D. collinear

Answer 1

The correct option is C

a right angled triangle

Explanation for correct option

Let the three sides of triangle are $AB,BC$ and $AC$

$$\begin{gathered} AB=2\hat{i}+\hat{j}-4\hat{k} \\ BC=3\hat{i}-2\hat{j}+\hat{k} \\ AC=\hat{i}-3\hat{j}+5\hat{k} \\ \left|AB\right|=\sqrt{2^2+1^2+{\left(-4\right)}^2}=\sqrt{4+1+16}=\sqrt{21} \\ \left|BC\right|=\sqrt{3^2+{\left(-2\right)}^2+1^2}=\sqrt{9+4+1}=\sqrt{14} \\ \left|AC\right|=\sqrt{1^2+3^2+5^2}=\sqrt{1+9+25}=\sqrt{35} \end{gathered}$$

Now if we square all the three magnitudes, then we get;

$$\begin{gathered} |AB{|}^2=21 \\ |BC{|}^2=14 \\ |AC{|}^2=35 \end{gathered}$$

As we can see,

$$\begin{gathered} |AB{|}^2+|BC{|}^2=|AC{|}^2 \\ \Rightarrow 21+14=35 \end{gathered}$$

The above condition is true for a right angled triangle

Hence option (C) is correct