Question

Show that the vectors $a=3\hat{i}-2\hat{j}+\hat{k},b=\hat{i}-3\hat{j}+5\hat{k}$ and $c=2\hat{i}+\hat{j}-4\hat{k}$ from a right angled triangle.

Answer 1

We have $a=(3\hat{i}-2\hat{j}+\hat{k})$ , $b=(\hat{i}-3\hat{j}-4\hat{k})$ and $c=(2\hat{i}+\hat{j}-4\hat{k})$

Since $b+c=a$ $\Rightarrow$ $(\hat{i}-3\hat{j}-4\hat{k})$ $+(2\hat{i}+\hat{j}-4\hat{k})$ $=(3\hat{i}-2\hat{j}+\hat{k})$

so, a, b and c formed a triangle.

Also, $\overrightarrow{a}.\overrightarrow{c}$ = $(3\hat{i}-2\hat{j}+\hat{k})$ $.(2\hat{i}+\hat{j}-4\hat{k})=(+6-2-4)=0$

So, $\overrightarrow{a}$ and $\overrightarrow{c}$ are perpendicular.

Therefore, $a,bandc$ form a right angled triangle.