# Minimum number of vectors needed in different planes for their resultant to be zero

I was doing some problems on physics when i saw a question that asked what is the minimum number of vectors needed in different planes for their resultant to be zero I thought about this and came to the conclusion that it should be three. ex : suppose one vector is $3\mathbf{\hat{i}}+ 3\mathbf{\hat{j}}$ in the $xy$ plane , another $-3\mathbf{\hat{i}} + 3\mathbf{\hat{k}}$ in the $xz$ plane and another $-3\mathbf{\hat{j}} -3\mathbf{\hat{k}}$ in the $yz$ plane .

So their resultants should be zero. But the answer is $4$. I don't understand why.

Please correct me if am making a mistake.

Thanks in advance.

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## 1 Answer

I think you may have some confusion about what you are being asked. To begin with in the three vectors you have described are coplanar i.e. they lie in the same plane, or to be specific the three vectors are linearly dependent.

You actually need one more vector to give a zero resultant.

See the plots below:  Both images are completely equivalent. The first being a plot of all three vectors and the second is a rotation showing they all lie in the same plane.