Question

What minimum number of non-zero non-collinear vectors is required to produce a zero vector?

Answer 1

Explanation:

A2A:If you have a vector v element of a vector space then you need -v to build a null vector. But v and -v are collinear. So you need more vectors.

Lets suppose there are two vectors v and w non collinear in this vector space, then we can add both vectors, to nullify this sum we need its inverse, so we get

(v + w) +(-(v+w)) = v + w + (-v) + (-w) = v + (-v) + w + (-w) = 0 + 0 = 0

choosese as third vector u;= -(v + w)

The sum is not collinear with v and w, because v and w are not collinear and so the inverse sum

So u,v,w are non collinear and sum up to zero vector.

More than three vectors are not necessary.