Answer :
To determine the minimum number of coplanar vectors with different magnitudes that can be added together to yield a resultant vector of zero, we can use some principles from vector geometry.
Step-by-Step Solution:
1. Understanding Coplanar Vectors:
- Coplanar vectors are vectors that lie in the same plane.
- For vectors to sum up to zero, they need to form a closed geometric shape such that their magnitudes and directions balance each other out.
2. Zero Resultant Vector:
- When the sum of all vectors in a system is zero, it means that there is no net motion or force in any direction within the plane.
- This condition implies that the vectors form a closed polygon, where each vector can be represented as a side of this polygon.
3. Minimum Number of Vectors:
- To form a closed shape using the least number of sides, three non-zero vectors must be considered.
- These vectors will sum to zero if they form a triangle because any side of a triangle can be seen as a vector that perfectly balances the other two sides through their magnitudes and directions.
4. Geometric Interpretation:
- A triangle is the simplest polygon that can be formed with straight-line segments. In this context, each side of the triangle is a vector.
- The conditions for these vectors to sum to zero are:
- They must have different magnitudes.
- They must be arranged such that the tip of one vector meets the tail of the next, forming a closed loop.
- The direction of these vectors should be such that they counterbalance each other.
Therefore, the minimum number of coplanar vectors with different magnitudes that can be added to give a zero resultant is three.
Step-by-Step Solution:
1. Understanding Coplanar Vectors:
- Coplanar vectors are vectors that lie in the same plane.
- For vectors to sum up to zero, they need to form a closed geometric shape such that their magnitudes and directions balance each other out.
2. Zero Resultant Vector:
- When the sum of all vectors in a system is zero, it means that there is no net motion or force in any direction within the plane.
- This condition implies that the vectors form a closed polygon, where each vector can be represented as a side of this polygon.
3. Minimum Number of Vectors:
- To form a closed shape using the least number of sides, three non-zero vectors must be considered.
- These vectors will sum to zero if they form a triangle because any side of a triangle can be seen as a vector that perfectly balances the other two sides through their magnitudes and directions.
4. Geometric Interpretation:
- A triangle is the simplest polygon that can be formed with straight-line segments. In this context, each side of the triangle is a vector.
- The conditions for these vectors to sum to zero are:
- They must have different magnitudes.
- They must be arranged such that the tip of one vector meets the tail of the next, forming a closed loop.
- The direction of these vectors should be such that they counterbalance each other.
Therefore, the minimum number of coplanar vectors with different magnitudes that can be added to give a zero resultant is three.
