The minimum number of vectors in different planes that can be added to give a zero resultant is 4.
The question, based on the provided reference, asks for the smallest number of vectors situated in distinct planes that can be combined vectorially to yield a zero resultant. This means the vectors, when added head-to-tail, form a closed polygon in space.
According to the reference:
The minimum number of vectors having different planes which can be added to give zero resultant is 4.07-Aug-2024
Let's break down why this minimum number is 4:
- One Vector: A single non-zero vector cannot have a zero resultant on its own.
- Two Vectors: Two vectors can sum to zero only if they are collinear (lie on the same line), equal in magnitude, and opposite in direction. If two non-zero vectors are in "different planes" in a way that they don't share a common line of action, they cannot sum to zero. Even if they are collinear, they would technically lie in the same plane (infinitely many planes, but they are coplanar).
- Three Vectors: Three vectors can sum to zero resultant only if they are coplanar. This means they must all lie within the same two-dimensional plane. If the three vectors are in different planes such that they cannot all be placed in the same plane, their vector sum cannot be zero. Imagine trying to close a triangle with sides not lying in the same flat surface – it's impossible.
- Four Vectors: With four or more vectors, it becomes possible for them to be located in different planes while still summing to a zero resultant. In three-dimensional space, the first three non-coplanar vectors define a certain resultant vector. The fourth vector can be positioned in a different plane and adjusted in magnitude and direction to exactly cancel out the resultant of the first three, thus achieving a total zero resultant. This requires moving out of a single plane into 3D space.
Practical Insight
Consider forces acting on an object in 3D space. If the object is in equilibrium, the vector sum of all forces acting on it is zero. While forces often act in multiple planes, a minimum of four forces might be required if they are oriented such that no three are coplanar, yet they still balance each other out.
This concept is fundamental in vector mechanics and physics, demonstrating how vectors in three dimensions behave differently than those confined to a plane.
In summary:
- 1 vector: Resultant cannot be zero (unless the vector is zero itself).
- 2 vectors: Can sum to zero only if collinear and equal/opposite.
- 3 vectors: Can sum to zero only if coplanar.
- 4 vectors: Can sum to zero even if no three are coplanar, allowing them to be in different planes and achieve a zero resultant in 3D space.
Therefore, the minimum number of vectors in different planes required for a zero resultant is 4, as stated in the reference.
