Yes if you put them "head to tail" and the head of the fourth one points to the tail of the first one the resultant is zero.
Take any three vectors in a plane which, when placed end-to-end form a triangle. The resultant of the three vectors will be zero.
Assuming you want non-zero vectors, two opposing vectors will give a resultant of zero.
No. For three vectors they must all lie in the same plane. Consider 2 vectors first. For them to resolve to zero, they must be in opposite direction and equal magnitude. So they will lie along the same line. For 3 vectors: take two of them. Any two vectors will lie in the same plane, and their resultant vector will also lie in that plane. Find the resultant of the first two vectors, and the third vector must be along the same line (equal magnitude, opposite direction), in order to result to zero. Since the third vector is along the same line as the resultant vector of the first two, then it must be in the same plane as the resultant of the first two. Therefore it lies in the same plane as the first two.
Yes, put the three vectors in a plane, with a separation of 120 degrees between each vector and each of the other vectors.
Two - if you add two vectors of equal magnitude but in opposite directions, the resultant vector is zero.
No. The tenth vector would have to be matched by one equal and opposite vector to yield a zero resultant, or by multiple vectors in the second plain collectively yielding a zero resultant for that plane. It would be possible, for example, for 8 vectors to be on the same plane and two on a different plane to give a zero resultant.
Take any three vectors in a plane which, when placed end-to-end form a triangle. The resultant of the three vectors will be zero.
Assuming you want non-zero vectors, two opposing vectors will give a resultant of zero.
No.
No. For three vectors they must all lie in the same plane. Consider 2 vectors first. For them to resolve to zero, they must be in opposite direction and equal magnitude. So they will lie along the same line. For 3 vectors: take two of them. Any two vectors will lie in the same plane, and their resultant vector will also lie in that plane. Find the resultant of the first two vectors, and the third vector must be along the same line (equal magnitude, opposite direction), in order to result to zero. Since the third vector is along the same line as the resultant vector of the first two, then it must be in the same plane as the resultant of the first two. Therefore it lies in the same plane as the first two.
Yes, put the three vectors in a plane, with a separation of 120 degrees between each vector and each of the other vectors.
Two - if you add two vectors of equal magnitude but in opposite directions, the resultant vector is zero.
Yes.
No.
The orientation of the three vectors that sum to zero must be coplanar, contained in the same common plane, including being contained in a common line in a plane.
Two vectors: no. Three vectors: yes.
No.