If position vectors of four points A, B, C, D are `hat(i)+hat(j)+hat(k), 2hat(i)+3hat(j), 3hat(i)+5hat(j)-2hat(k), -hat(j)+hat(k)` respectively, then `overline(AB) and overline(CD)` are related as

Options

A. perpendicualr
B. parallel
C. independent
D. `overline(AB)*overline(CD)=6`

Answer

To solve the problem, we need to find the vectors \( \overline{AB} \) and \( \overline{CD} \) using the given position vectors of points A, B, C, and D.

### Step 1: Identify the position vectors
The position vectors of the points are given as:
- \( \vec{A} = \hat{i} + \hat{j} + \hat{k} \)
- \( \vec{B} = 2\hat{i} + 3\hat{j} \)
- \( \vec{C} = 3\hat{i} + 5\hat{j} - 2\hat{k} \)
- \( \vec{D} = -\hat{j} + \hat{k} \)

### Step 2: Calculate the vector \( \overline{AB} \)
The vector \( \overline{AB} \) is calculated as:
\[ \overline{AB} = \vec{B} - \vec{A} \]
Substituting the position vectors:
\[ \overline{AB} = (2\hat{i} + 3\hat{j}) - (\hat{i} + \hat{j} + \hat{k}) \]
Now, perform the subtraction:
\[ \overline{AB} = (2\hat{i} - \hat{i}) + (3\hat{j} - \hat{j}) + (0 - \hat{k}) \]
\[ \overline{AB} = \hat{i} + 2\hat{j} - \hat{k} \]

### Step 3: Calculate the vector \( \overline{CD} \)
The vector \( \overline{CD} \) is calculated as:
\[ \overline{CD} = \vec{D} - \vec{C} \]
Substituting the position vectors:
\[ \overline{CD} = (-\hat{j} + \hat{k}) - (3\hat{i} + 5\hat{j} - 2\hat{k}) \]
Now, perform the subtraction:
\[ \overline{CD} = (0 - 3\hat{i}) + (-\hat{j} - 5\hat{j}) + (\hat{k} + 2\hat{k}) \]
\[ \overline{CD} = -3\hat{i} - 6\hat{j} + 3\hat{k} \]

### Step 4: Determine the relationship between \( \overline{AB} \) and \( \overline{CD} \)
We can express \( \overline{CD} \) in terms of \( \overline{AB} \):
\[ \overline{CD} = -3(\hat{i} + 2\hat{j} - \hat{k}) = -3\overline{AB} \]
This shows that \( \overline{CD} \) is a scalar multiple of \( \overline{AB} \).

### Conclusion
Since \( \overline{CD} \) can be expressed as a scalar multiple of \( \overline{AB} \), we conclude that the two vectors are parallel.

Thus, the correct answer is:
**Option B: Parallel**