How many minimum numbers of a coplanar vector having different magntidues can be added to give zero resultant

Options

A. 2
B. 3
C. 4
D. 5

Answer

To solve the problem of how many minimum coplanar vectors with different magnitudes can be added to give a zero resultant, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**: We need to find the minimum number of coplanar vectors that can be added together to result in a zero vector. The vectors must have different magnitudes.

2. **Using the Triangle Law of Vector Addition**: According to the triangle law of vector addition, for two vectors to add up to zero, they must be equal in magnitude but opposite in direction. However, since the problem specifies that the vectors must have different magnitudes, we cannot use just two vectors.

3. **Considering Three Vectors**: Let's consider three vectors, \( \vec{A} \), \( \vec{B} \), and \( \vec{C} \). For these vectors to sum to zero, they must form a closed triangle when arranged head-to-tail.

4. **Arranging the Vectors**: We can arrange the three vectors such that:
- Vector \( \vec{A} \) points in one direction.
- Vector \( \vec{B} \) points in a different direction.
- Vector \( \vec{C} \) connects the head of \( \vec{B} \) back to the tail of \( \vec{A} \).

5. **Condition for Zero Resultant**: For the resultant of these three vectors to be zero, the vectors must satisfy the condition:
\[ \vec{A} + \vec{B} + \vec{C} = 0 \]
This implies that the vectors must be arranged in such a way that they form a closed loop.

6. **Conclusion**: Since we have established that two vectors cannot satisfy the condition due to the requirement of different magnitudes, and three vectors can be arranged to form a closed triangle, the minimum number of coplanar vectors with different magnitudes that can be added to give a zero resultant is **three**.

### Final Answer:
The minimum number of coplanar vectors with different magnitudes that can be added to give a zero resultant is **3**.

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