Introduction:
In the context of vectors, coplanar vectors lie in the same plane, while non-coplanar vectors do not lie in the same plane. The resultant of vectors is the vector formed by adding or subtracting the individual vectors.

Explanation:
It is not possible for three non-coplanar vectors to give a zero resultant. This can be understood by considering the properties of vectors and their addition.

Property 1: Vectors in a Plane:
When three vectors lie in the same plane, they are called coplanar vectors. In this case, the sum of the three vectors can be zero if they form a closed triangle. However, if they do not form a closed triangle, the sum of the vectors will not be zero.

Property 2: Non-Coplanar Vectors:
When three vectors do not lie in the same plane, they are called non-coplanar vectors. In this case, it is not possible for the sum of the vectors to be zero.

Explanation:
To understand why the sum of three non-coplanar vectors cannot be zero, let's consider an example.

Suppose we have three vectors A, B, and C. If these vectors are non-coplanar, it means they do not lie in the same plane.

When we add vectors A, B, and C, we perform vector addition by adding the corresponding components of the vectors. The sum of the vectors is given by:
A + B + C = (Ax + Bx + Cx)i + (Ay + By + Cy)j + (Az + Bz + Cz)k

For the sum of vectors A + B + C to be zero, all the components of the resultant vector must be zero. This implies that:
Ax + Bx + Cx = 0
Ay + By + Cy = 0
Az + Bz + Cz = 0

However, since the vectors A, B, and C are non-coplanar, they do not lie in the same plane. Therefore, it is not possible for their components to add up to zero simultaneously.

Conclusion:
In conclusion, three non-coplanar vectors cannot give a zero resultant. This is due to the fact that their components cannot add up to zero simultaneously as they do not lie in the same plane.