The position vectors of points A, B , C and D are `vec(A) = 3hat(i) + 4hat(j) + 5hat(k), vec(B) = 4hat(i)+5hat(j) + 6hat(k), vec(C )= 7hat(i) + 9hat(j) + 3hat(k)` and `vec(D) = 4hat(i) + 6hat(j)` then the displacement vectors `bar(AB)` and `bar(CD)` are :

Options

A. perpendicular
B. Parallel
C. Antiparallel
D. Inclined at an angle of `60^(@)`

Answer

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

### Step 1: Write down the position vectors
We have the following position vectors:
- \(\vec{A} = 3\hat{i} + 4\hat{j} + 5\hat{k}\)
- \(\vec{B} = 4\hat{i} + 5\hat{j} + 6\hat{k}\)
- \(\vec{C} = 7\hat{i} + 9\hat{j} + 3\hat{k}\)
- \(\vec{D} = 4\hat{i} + 6\hat{j}\)

### Step 2: Calculate the displacement vector \(\vec{AB}\)
The displacement vector \(\vec{AB}\) is given by:
\[ \vec{AB} = \vec{B} - \vec{A} \]
Substituting the position vectors:
\[ \vec{AB} = (4\hat{i} + 5\hat{j} + 6\hat{k}) - (3\hat{i} + 4\hat{j} + 5\hat{k}) \]
Now, perform the subtraction:
\[ \vec{AB} = (4 - 3)\hat{i} + (5 - 4)\hat{j} + (6 - 5)\hat{k} \]
This simplifies to:
\[ \vec{AB} = 1\hat{i} + 1\hat{j} + 1\hat{k} = \hat{i} + \hat{j} + \hat{k} \]

### Step 3: Calculate the displacement vector \(\vec{CD}\)
The displacement vector \(\vec{CD}\) is given by:
\[ \vec{CD} = \vec{D} - \vec{C} \]
Substituting the position vectors:
\[ \vec{CD} = (4\hat{i} + 6\hat{j}) - (7\hat{i} + 9\hat{j} + 3\hat{k}) \]
Now, perform the subtraction:
\[ \vec{CD} = (4 - 7)\hat{i} + (6 - 9)\hat{j} + (0 - 3)\hat{k} \]
This simplifies to:
\[ \vec{CD} = -3\hat{i} - 3\hat{j} - 3\hat{k} = -3(\hat{i} + \hat{j} + \hat{k}) \]

### Step 4: Final results
Thus, the displacement vectors are:
\[ \vec{AB} = \hat{i} + \hat{j} + \hat{k} \]
\[ \vec{CD} = -3(\hat{i} + \hat{j} + \hat{k}) \]