If `overset(rarr)B=noverset(rarr)A` and `overset(rarr)A` is antiparallel with `overset(rarr)B`, then n is :

Options

A. a vector
B. negative and dimensions less
C. a scalar
D. a negative scalar

Explanation

To solve the problem, we need to analyze the relationship between the vectors \(\overset{\rarr}{B}\) and \(\overset{\rarr}{A}\) given that \(\overset{\rarr}{B} = n \overset{\rarr}{A}\) and that \(\overset{\rarr}{A}\) is antiparallel to \(\overset{\rarr}{B}\).

### Step-by-Step Solution:

1. **Understanding Antiparallel Vectors**:
- Two vectors are said to be antiparallel if they point in opposite directions. This means that the angle between them is \(180^\circ\).
- If \(\overset{\rarr}{A}\) is in one direction, then \(\overset{\rarr}{B}\) must be in the exact opposite direction.

2. **Expressing \(\overset{\rarr}{B}\)**:
- Given that \(\overset{\rarr}{B} = n \overset{\rarr}{A}\), we can express \(\overset{\rarr}{B}\) in terms of \(\overset{\rarr}{A}\).
- Since \(\overset{\rarr}{B}\) is antiparallel to \(\overset{\rarr}{A}\), we can write:
\[ \overset{\rarr}{B} = -k \overset{\rarr}{A} \]
where \(k\) is a positive scalar.

3. **Equating the Two Expressions**:
- From the above, we have:
\[ n \overset{\rarr}{A} = -k \overset{\rarr}{A} \]
- Since \(\overset{\rarr}{A}\) is non-zero, we can divide both sides by \(\overset{\rarr}{A}\):
\[ n = -k \]
- This implies that \(n\) must be a negative scalar.

4. **Identifying the Nature of \(n\)**:
- Since \(k\) is a positive scalar, \(n\) must be negative. Therefore, \(n\) is a negative scalar quantity.
- Additionally, \(n\) is dimensionless because it is a ratio of two vectors of the same dimension (both are vectors).

5. **Conclusion**:
- The value of \(n\) is a negative scalar quantity, which means it can take values like \(-1\), \(-2\), \(-3\), etc.

### Final Answer:
- \(n\) is a negative scalar.