Let$\ \vec a, \vec b, \vec c \ $ be noncollinear vectors. Show that the necesssary and sufficient condition for the existence of a triangle $ABC$ with the properties $\vec {BC}=\vec a ,\vec {CA}= \vec b , \vec {AB}=\vec c \ \ $is$ \ \vec a × \vec b = \vec b × \vec c = \vec c × \vec a $. From the equalities of the norms deduce the law of sines.
Can somebody give me some tips, please? I don't know what condition for existance of a triangle to use.
$$\vec{BC}+\vec{CA}+\vec{AB}=\vec{BB}=0$$ You can write this as $$\vec a+\vec b+\vec c=0$$ Now multiply this equation by $\vec c$ (vector product, not scalar) $$\vec a\times\vec c+\vec b\times\vec c+\vec c\times\vec c=0$$ Use properties of the vector product.