Question

The position vectors of the points $A,B$ and $C$ are $\hat{i}+\hat{j}+\hat{k},\hat{i}+5\hat{j}-k$ and $2\hat{i}+3\hat{j}+5k$ respectively. The greatest angle of the triangle $ABC$ is

• A. ${90}^o$
• B. ${135}^o$
• C. ${\cos }^{-1}(\frac{2}{3})$
• D. ${\cos }^{-1}\left(\frac{5}{7}\right)$

Answer 1

The correct option is A ${90}^o$

$a=|BC|=|1\overline{i}-2\overline{j}+6\overline{k}|=\sqrt{41}$
$b=|AC|=|1\overline{i}+2\overline{j}+4\overline{k}|=\sqrt{21}$
$c=|AB|=|4\overline{i}-2\overline{j}|=\sqrt{20}$
Looking at the figures, we come to know that the triangle is a right angled triangle with right angle at $A$.
Thus, its the greatest angle.