How many minimum numbers of non zero vectors in different planes can be added to give zero resultant.

Options

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

Answer

To find the minimum number of non-zero vectors in different planes that can be added to give a zero resultant, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Vectors in Different Planes**:
- We need to consider vectors that are not confined to a single plane. This means that each vector must lie in a different plane.

2. **Starting with Two Vectors**:
- Let's denote the first vector as **A** and the second vector as **B**. When we add these two vectors, they will produce a resultant vector **R** that lies in the plane formed by **A** and **B**.

3. **Resultant of Two Vectors**:
- The resultant vector **R** can be expressed as:
\[ \mathbf{R} = \mathbf{A} + \mathbf{B} \]
- Since we want a zero resultant, we need to counteract this resultant.

4. **Adding a Third Vector**:
- To achieve a zero resultant, we can introduce a third vector **C** that is equal in magnitude but opposite in direction to **R**:
\[ \mathbf{C} = -\mathbf{R} \]
- However, **A**, **B**, and **C** are all in the same plane, which does not satisfy our requirement of having vectors in different planes.

5. **Introducing More Vectors**:
- To satisfy the condition of different planes, we need to introduce additional vectors. Let's denote these additional vectors as **D** and **E**.
- We can arrange **C**, **D**, and **E** such that they lie in different planes.

6. **Balancing the Resultant**:
- We can set the condition that the resultant of vectors **C**, **D**, and **E** must equal **-R**:
\[ \mathbf{C} + \mathbf{D} + \mathbf{E} = -\mathbf{R} \]
- Thus, we can express the overall condition as:
\[ \mathbf{A} + \mathbf{B} + \mathbf{C} + \mathbf{D} + \mathbf{E} = 0 \]

7. **Conclusion**:
- Therefore, the minimum number of non-zero vectors in different planes that can be added to give a zero resultant is **4**. These vectors are **A**, **B**, **C**, and **D** (or **E**), where **C** and **D** are in different planes from **A** and **B**.

### Final Answer:
The minimum number of non-zero vectors in different planes that can be added to give a zero resultant is **4**.