Question

If $\overline{a}$, $\overline{b}$, $\overline{c}$ be three unit vectors, such that $\overline{a}+\overline{b}+\overline{c}$ is also a unit vector and ${\alpha }_1$, ${\alpha }_2$, ${\alpha }_3$ be the angles between $\overline{a}$ and $\overline{b}$, $\overline{b}$ and $\overline{c}$, $\overline{c}$ and $\overline{a}$ respectively, then ${\alpha }_1$, ${\alpha }_2$, ${\alpha }_3$

• A. All are acute angles
• B. All are right angles
• C. Has at least one among them obtuse
• D. None of these

Answer 1

Answer: (C)

Has at least one among them obtuse

${\left|(\overline{a}+\overline{b}+\overline{c})\right|}^2=1$
$={\left|\overline{a}\right|}^2+{\left|\overline{b}\right|}^2+{\left|\overline{c}\right|}^2+2\left|\overline{a}\right|\left|\overline{b}\right|\cos {\alpha }_1+2\left|\overline{b}\right|\left|\overline{c}\right|\cos {\alpha }_2+2\left|\overline{c}\right|\left|\overline{a}\right|\cos {\alpha }_3=1$
$\Rightarrow$ $\cos {\alpha }_1+\cos {\alpha }_2+\cos {\alpha }_3=-1$
$\therefore$ Atleast one of the above terms is negative.
$\therefore$ At least one of ${\alpha }_1$, ${\alpha }_2$, ${\alpha }_3$ should be obtuse.