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 `vec(AB)` and `vec(CD)` are :

Options

A. perpendicular
B. parallel
C. anti-parallel
D. inclined at an angle of `60^(@)`

Answer

To find the displacement vectors \(\vec{AB}\) and \(\vec{CD}\) given the position vectors of points A, B, C, and D, we will follow these steps:

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

### Step 2: Calculate the displacement vector \(\vec{AB}\)
The displacement vector \(\vec{AB}\) is calculated as:
\[ \vec{AB} = \vec{B} - \vec{A} \]
Substituting the values:
\[ \vec{AB} = (4\hat{i} + 5\hat{j} + 6\hat{k}) - (3\hat{i} + 4\hat{j} + 5\hat{k}) \]
Calculating component-wise:
\[ \vec{AB} = (4 - 3)\hat{i} + (5 - 4)\hat{j} + (6 - 5)\hat{k} \]
\[ \vec{AB} = 1\hat{i} + 1\hat{j} + 1\hat{k} \]

### Step 3: Calculate the displacement vector \(\vec{CD}\)
The displacement vector \(\vec{CD}\) is calculated as:
\[ \vec{CD} = \vec{D} - \vec{C} \]
Substituting the values:
\[ \vec{CD} = (4\hat{i} + 6\hat{j}) - (7\hat{i} + 9\hat{j} + 3\hat{k}) \]
Calculating component-wise:
\[ \vec{CD} = (4 - 7)\hat{i} + (6 - 9)\hat{j} + (0 - 3)\hat{k} \]
\[ \vec{CD} = -3\hat{i} - 3\hat{j} - 3\hat{k} \]

### Step 4: Summary of the displacement vectors
Thus, we have:
\[ \vec{AB} = 1\hat{i} + 1\hat{j} + 1\hat{k} \]
\[ \vec{CD} = -3\hat{i} - 3\hat{j} - 3\hat{k} \]

### Step 5: Determine the relationship between \(\vec{AB}\) and \(\vec{CD}\)
To determine the relationship (whether they are parallel, anti-parallel, or perpendicular), we can calculate the dot product:
\[ \vec{AB} \cdot \vec{CD} = (1)(-3) + (1)(-3) + (1)(-3) = -3 - 3 - 3 = -9 \]

Next, we check the magnitudes:
\[ |\vec{AB}| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3} \]
\[ |\vec{CD}| = \sqrt{(-3)^2 + (-3)^2 + (-3)^2} = \sqrt{27} = 3\sqrt{3} \]

### Step 6: Check for anti-parallel condition
For two vectors to be anti-parallel, the dot product should equal the negative product of their magnitudes:
\[ \vec{AB} \cdot \vec{CD} = -|\vec{AB}| \cdot |\vec{CD}| \]
Calculating:
\[ -|\vec{AB}| \cdot |\vec{CD}| = -(\sqrt{3})(3\sqrt{3}) = -9 \]
Since both sides are equal, we conclude that \(\vec{AB}\) and \(\vec{CD}\) are anti-parallel.

### Final Answer
\[ \vec{AB} = 1\hat{i} + 1\hat{j} + 1\hat{k}, \quad \vec{CD} = -3\hat{i} - 3\hat{j} - 3\hat{k} \]
The vectors \(\vec{AB}\) and \(\vec{CD}\) are anti-parallel.