If ABCD is a parallelogram and the position vectors of A,B and C are `hati+3hatj+5hatk, hati+hatj+hatk and 7 hati+7hatj+7hatk`, then the poisitionn vector of D will be

Options

A. `7hati+5hatj+3hatk`
B. `7hati+9hatj+11hatk`
C. `9hati+11hatj+13hatk`
D. `8hati+8hatj+8hatk`

Explanation

To find the position vector of point D in the parallelogram ABCD, we can use the properties of vectors and the relationships between the vertices of the parallelogram.

Given:
- Position vector of A: **A** = \( \hat{i} + 3\hat{j} + 5\hat{k} \)
- Position vector of B: **B** = \( \hat{i} + \hat{j} + \hat{k} \)
- Position vector of C: **C** = \( 7\hat{i} + 7\hat{j} + 7\hat{k} \)

### Step 1: Understand the properties of a parallelogram
In a parallelogram, the opposite sides are equal in vector form. Therefore, we have:
\[ \overrightarrow{AB} = \overrightarrow{DC} \]

### Step 2: Calculate the vector **AB**
The vector **AB** can be calculated as:
\[ \overrightarrow{AB} = \overrightarrow{B} - \overrightarrow{A} = (\hat{i} + \hat{j} + \hat{k}) - (\hat{i} + 3\hat{j} + 5\hat{k}) \]
\[ = \hat{i} + \hat{j} + \hat{k} - \hat{i} - 3\hat{j} - 5\hat{k} \]
\[ = 0\hat{i} - 2\hat{j} - 4\hat{k} \]
Thus,
\[ \overrightarrow{AB} = -2\hat{j} - 4\hat{k} \]

### Step 3: Set up the equation for vector **DC**
Since **AB** is equal to **DC**, we can express **DC** as:
\[ \overrightarrow{DC} = \overrightarrow{C} - \overrightarrow{D} \]
Let the position vector of D be \( \overrightarrow{D} = x\hat{i} + y\hat{j} + z\hat{k} \). Then:
\[ \overrightarrow{DC} = (7\hat{i} + 7\hat{j} + 7\hat{k}) - (x\hat{i} + y\hat{j} + z\hat{k}) \]
\[ = (7 - x)\hat{i} + (7 - y)\hat{j} + (7 - z)\hat{k} \]

### Step 4: Set the vectors equal
Since \( \overrightarrow{AB} = \overrightarrow{DC} \), we have:
\[ 0\hat{i} - 2\hat{j} - 4\hat{k} = (7 - x)\hat{i} + (7 - y)\hat{j} + (7 - z)\hat{k} \]

### Step 5: Equate the coefficients of \( \hat{i}, \hat{j}, \hat{k} \)
From the equation, we can equate the coefficients:
1. For \( \hat{i} \):
\[ 0 = 7 - x \implies x = 7 \]
2. For \( \hat{j} \):
\[ -2 = 7 - y \implies y = 9 \]
3. For \( \hat{k} \):
\[ -4 = 7 - z \implies z = 11 \]

### Step 6: Write the position vector of D
Now we can write the position vector of D:
\[ \overrightarrow{D} = 7\hat{i} + 9\hat{j} + 11\hat{k} \]

### Final Answer
Thus, the position vector of D is:
\[ \overrightarrow{D} = 7\hat{i} + 9\hat{j} + 11\hat{k} \]