I was solving a question in which I had to prove three vectors $\vec a, \vec b, \vec c$ form a right angled triangle.
What I did was find the magnitude of the three vectors and showed that they followed Pythagoras theorem.
Now when I saw the solution to cross check, it said, in the first step, $\vec b+ \vec c = \vec a$ hence they are coplanar.
Is this true? I thought if the scalar triple product of three vectors is zero then they are coplanar.
Also, is there even a need to prove the vectors coplanar??
Best Answer
Yeah, that's true. Of course, the scalar triple product being $0$ means that the vectors are coplanar. However, in this case, we need not use that. Think about it, the sum of two vectors always stays in the plane defined by them. Hence, in this case, $\vec{a}$ is in the same plane as $\vec{b}$ and $\vec{c}$ and hence, they are coplanar. Now, your method of checking scalar product would've also worked here. But, this is just quicker and smarter.
Also, there indeed is a need to check this. Because, the vectors might have compatible lengths and yet be colinear or something else, barring them from forming a triangle at all.