Browse Questions

If $\overrightarrow a,\overrightarrow b,\overrightarrow c$ are unit vectors and $\overrightarrow a+\overrightarrow b+\overrightarrow c$ is also a unit vector and $\alpha,\beta,\gamma$ are angles between $\overrightarrow a\:and\:\overrightarrow b,\:\:\overrightarrow b\:and\:\overrightarrow c\:\:\overrightarrow c\:and\:\overrightarrow a$ respectively, then among the three angles,

$(a)\:All\:the\:angles\:are\:acute\:\:\qquad\:\:(b)\:\:All\:are\:rt.\:angles.\:\:\qquad\:\:(c)\:\:At\:least\:one\:is\:obtuse.\:\:\qquad\:\:(d)\:\:None\:of\:these.$

Toolbox:
• $\overrightarrow a.\overrightarrow b=|\overrightarrow a||\overrightarrow b|cos\theta$
Given: $|\overrightarrow a|=|\overrightarrow b|=|\overrightarrow c|=|\overrightarrow a+\overrightarrow b+\overrightarrow c|=1$
$|\overrightarrow a+\overrightarrow b+\overrightarrow c|^2=|\overrightarrow a|^2+|\overrightarrow b|^2+|\overrightarrow c|^2+2(\overrightarrow a.\overrightarrow b+\overrightarrow b.\overrightarrow c+\overrightarrow c.\overrightarrow a)$
$\Rightarrow\:1=3+2(cos\alpha+cos\beta+cos\gamma)$
$\Rightarrow\:cos\alpha+cos\beta+cos\gamma=-1$
$\Rightarrow\:$ atleast one out of $cos\alpha,\:cos\beta,\:cos\gamma$ should be $-ve$.
$\Rightarrow\:$ At least one out of the three angles should be obtuse.