State whether the following relations are true or false
(1)`vec(AB)=vec(BA)`
(2)`vec(AB)=-vec(BA)`
(3)`|vec(AB)|=|vec(BA)|`
(4)`|vec(AB)|=|-vec(AB)|`
(5)`hatj=hatk`
(6)`|hatj|=|hatk|`

Answer

Let's analyze each of the statements one by one to determine whether they are true or false.

### Step-by-Step Solution:

1. **Statement (1):** `vec(AB) = vec(BA)`
- **Analysis:** The vector `vec(AB)` points from point A to point B, while `vec(BA)` points from point B to point A. They have the same magnitude but opposite directions.
- **Conclusion:** This statement is **False**.

2. **Statement (2):** `vec(AB) = -vec(BA)`
- **Analysis:** Since `vec(AB)` points from A to B and `vec(BA)` points from B to A, we can say that `vec(AB)` is indeed the negative of `vec(BA)`. This means that if you reverse the direction of `vec(BA)`, you get `vec(AB)`.
- **Conclusion:** This statement is **True**.

3. **Statement (3):** `|vec(AB)| = |vec(BA)|`
- **Analysis:** Both vectors `vec(AB)` and `vec(BA)` have the same magnitude because they represent the same distance between points A and B, regardless of direction.
- **Conclusion:** This statement is **True**.

4. **Statement (4):** `|vec(AB)| = |-vec(AB)|`
- **Analysis:** The magnitude of a vector is always a non-negative quantity. Therefore, the magnitude of `vec(AB)` is equal to the magnitude of `-vec(AB)`, since both represent the same distance.
- **Conclusion:** This statement is **True**.

5. **Statement (5):** `hatj = hatk`
- **Analysis:** The unit vector `hatj` represents the y-axis direction, while `hatk` represents the z-axis direction. These are different directions.
- **Conclusion:** This statement is **False**.

6. **Statement (6):** `|hatj| = |hatk|`
- **Analysis:** Both `hatj` and `hatk` are unit vectors, meaning their magnitudes are both equal to 1.
- **Conclusion:** This statement is **True**.

### Final Summary of Statements:
1. False
2. True
3. True
4. True
5. False
6. True