How do we know if a third vector is on the plane of the first and second vector? I have to add its components and if it gives me zero,then is it on the plane?
Suppose that the vectors have three components.
Thank you.

Use the fact that a plane is a two dimensional object. That means that the size of the base is two (you can describe any vector in that plane as a linear combination of the vectors in the base). Now create a matrix with the components of those three vectors as rows (or columns). If the determinant is not zero, then the vectors are independent, so they span a three dimensional space, not two. You need to have the determinant equal to $0$
