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I have a basic homework vectors question which I can't figure. let's say I have three 3d vectors: $$\vec{a}=2\mathbf{i}-2\mathbf{j}-\mathbf{k}\\ \vec{b}=6\mathbf{i}-3\mathbf{j}+2\mathbf{k}\\ \vec{c}=4\mathbf{i}-\mathbf{j}+3\mathbf{k} $$ How can I tell if the 3 vectors are creating a triangle? I was thinking about 2 things. first, I can check the angle between every 2 of them using the dot (scalar) method and sum the angles to see if it's $180^\circ$...it's not. but Iv'e noticed that vectors $\vec{a}+\vec{c}=\vec{b}$ so $\vec{b}$ is closing a triangle if I think about it this way... What is actually true? Thanks!

Harish Chandra Rajpoot
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user3921
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    Would [math.se] be a better home for this question? – Qmechanic Nov 10 '13 at 11:36
  • Bogdans answer is of course correct, but since you tried to verify the angle-sum and somehow failed, tell us what you calculated. Because you should end up with Pi – Nephente Nov 10 '13 at 11:55

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If the vectors form any closed polygon then traversing them all should bring you back where you started:

$$ \vec{a} + \vec{b} + \vec{c} = 0 $$

In this case the vectors do not form a triangle, though they do if you replace $\vec{b}$ by $-\vec{b}$ i.e. make $\vec{b}$ point in the other direction.

John Rennie
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First thing to consider doing is to check (as they are 3D vectors) whether they are coplanar vectors or not. For this, take the scalar triple product of these three vectors, if its zero, which it certainly is in this example, then they are coplanar. After coplanarity if conformed, then you can either just see if the addition of any two leads to the third, which in this case it doesn't , OR you can calculate the angles between each vectors (by dot product) and add them to see if it's sum is 180° or not.

Zeeshan Shaikh
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The vectors form a triangle because, as you've mentioned, $~ \vec{a} + \vec{c}~=~\vec{b}$ .
EDIT: I'm sorry for the previous statement, but they do not form a triangle, actually. You were on the right way: you need to check if the sum of angles is $\pi$ and to check if two of the vectors can actually form another given vector (in the case this fulfils).
It appears that vector $\vec{b}$ is between vectors $\vec{a}$ and $\vec{c}$. You'll see it by calculating arcs between every possible pair of your vectors.

Bohdan
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