Which of the following is correct relation between an arbitrary vector A and null vector O ?

Options

A. `A + O + A xx O = A`
B. `A + O + A xx O != A`
C. `A + O + A xx O = O`
D. None of these

Answer

To solve the question regarding the relationship between an arbitrary vector \( \mathbf{A} \) and the null vector \( \mathbf{O} \), we need to analyze the properties of these vectors step by step.

### Step-by-Step Solution:

1. **Understanding the Null Vector**:
The null vector, denoted as \( \mathbf{O} \), is defined as:
\[ \mathbf{O} = 0 \hat{i} + 0 \hat{j} + 0 \hat{k} \]
This means that it has no magnitude and no direction.

**Hint**: Remember that the null vector is essentially the vector with all components equal to zero.

2. **Addition of Vectors**:
One of the fundamental properties of vectors is that when any vector is added to the null vector, the result is the original vector itself. Mathematically, this can be expressed as:
\[ \mathbf{A} + \mathbf{O} = \mathbf{A} \]

**Hint**: Think of the null vector as having no effect when added to another vector.

3. **Multiplication of Vectors**:
When we consider the multiplication of a vector with the null vector, we can look at both the dot product and the cross product:
- **Dot Product**:
\[ \mathbf{A} \cdot \mathbf{O} = 0 \]
This indicates that the dot product of any vector with the null vector is zero.
- **Cross Product**:
\[ \mathbf{A} \times \mathbf{O} = \mathbf{O} \]
The cross product of any vector with the null vector is also the null vector.

**Hint**: Remember that the dot product measures the extent to which two vectors point in the same direction, and since the null vector has no direction, the result is zero.

4. **Analyzing the Given Options**:
Now, we need to evaluate the given options based on the properties we discussed:
- **Option 1**: \( \mathbf{A} + \mathbf{O} + \mathbf{A} \cdot \mathbf{O} \)
- Using the properties:
\[ \mathbf{A} + \mathbf{O} + 0 = \mathbf{A} \]
- This option is correct.
- **Option 2**: \( \mathbf{A} + \mathbf{O} + \mathbf{A} \cdot \mathbf{O} \neq \mathbf{A} \)
- This statement is incorrect since we already established that it equals \( \mathbf{A} \).
- **Option 3**: Similar to Option 2, it will also lead to \( \mathbf{A} \) and not \( \mathbf{O} \).
- **Option 4**: This will also lead to \( \mathbf{A} \) and not \( \mathbf{O} \).

**Hint**: Check each option carefully against the properties of the null vector.

5. **Conclusion**:
The only correct relation between the arbitrary vector \( \mathbf{A} \) and the null vector \( \mathbf{O} \) is found in Option 1.

### Final Answer:
The correct relation is given by **Option 1**: \( \mathbf{A} + \mathbf{O} + \mathbf{A} \cdot \mathbf{O} = \mathbf{A} \).