Question: (I made it up) Given three vectors $\vec {AB}, \vec {BC},\vec {CA}$; what is the complete method to prove that they form a triangle.
I know about the triangle law of addition, and that if $\vec {AB}+ \vec {BC}+\vec {CA}=0$, then these vectors may form a triangle.
However, my text book gives one example of: $\vec {AB}=3i-j-2k; \vec {BC}=6i-2j-4k;\vec {CA}=9i-3j-6k$ where it notes that even though these three add up to zero, yet they do not form a triangle.
My textbook doesn't give us a list of "checks" that need to be made on a given set of three vectors so as to check if they form a valid triangle.
I don't want to fall prey to examples of vectors as above, so I require one such list.
I know 2-3 questions have been asked before on this topic but they're all having scattered theory and mine is an attempt to clear my doubts while unifying all steps together.
Best Answer
To check whether $3$ vectors form a non-degenerate triangle, i.e. a triangle with positive area, you only need to check if the sum of them is $0$ and the are not colinear.
$3$ vectors that sums $0$ can be written as $\vec{AB} + \vec{BC} + \vec{CA} = 0$ so $\vec{AB} + \vec{BC} = -\vec{CA}$. The are not linearly independent so they are coplanar.