Question

Show that the points A, B, C with position vectors $2\hat{i}-\hat{j}+\hat{k},\hat{i}-3\hat{j}-5\hat{k}$ and $3\hat{i}-4\hat{j}-4\hat{k}$ respectively, are the vertices of a right angled triangle. Hence find the area of the triangle.

Answer 1

Distance between $A$ and $B$ is $\sqrt{{(2-1)}^2+{(-1+3)}^2+{(1+5)}^2}=\sqrt{41}$
Distance between $B$ and $C$ is $\sqrt{{(1-3)}^2+{(-3+4)}^2+{(-5+4)}^2}=\sqrt{6}$

Distance between $C$ and $A$ is $\sqrt{{(3-2)}^2+{(-4+1)}^2+{(-4-1)}^2}=\sqrt{35}$

We can clearly see that ${BC}^2+{AC}^2={AB}^2$

By pythogorean theorem ABC form a right angled triangle with right angle at C

Area of triangle $=\frac{1}{2}\times BC\times AC$

$\Rightarrow \frac{1}{2}\times \sqrt{6}\times \sqrt{35}$

$\Rightarrow \frac{\sqrt{210}}{2}$