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 poisition 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 in a parallelogram. The position vectors of points A, B, and C are given as follows:

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

Let the position vector of point D be \( \vec{D} = x\hat{i} + y\hat{j} + z\hat{k} \).

### Step 1: Use the property of the parallelogram
In a parallelogram, the vector \( \vec{AB} \) is equal to the vector \( \vec{DC} \). This can be expressed as:
\[ \vec{AB} = \vec{B} - \vec{A} = \vec{C} - \vec{D} = \vec{DC} \]

### Step 2: Calculate \( \vec{AB} \)
Now, we calculate \( \vec{AB} \):
\[ \vec{AB} = \vec{B} - \vec{A} = (\hat{i} + \hat{j} + \hat{k}) - (\hat{i} + 3\hat{j} + 5\hat{k}) \]
\[ = \hat{i} + \hat{j} + \hat{k} - \hat{i} - 3\hat{j} - 5\hat{k} \]
\[ = 0\hat{i} - 2\hat{j} - 4\hat{k} \]
\[ = -2\hat{j} - 4\hat{k} \]

### Step 3: Set up the equation for \( \vec{DC} \)
Now, we can express \( \vec{DC} \):
\[ \vec{DC} = \vec{C} - \vec{D} = (7\hat{i} + 7\hat{j} + 7\hat{k}) - (x\hat{i} + y\hat{j} + z\hat{k}) \]
\[ = (7 - x)\hat{i} + (7 - y)\hat{j} + (7 - z)\hat{k} \]

### Step 4: Set \( \vec{AB} = \vec{DC} \)
Since \( \vec{AB} = \vec{DC} \), we can equate the two expressions:
\[ 0\hat{i} - 2\hat{j} - 4\hat{k} = (7 - x)\hat{i} + (7 - y)\hat{j} + (7 - z)\hat{k} \]

### Step 5: Compare coefficients
From the equation above, we can compare the coefficients of \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \):

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, substituting the values of \( x \), \( y \), and \( z \) back into the position vector of D:
\[ \vec{D} = 7\hat{i} + 9\hat{j} + 11\hat{k} \]

### Final Answer
The position vector of point D is:
\[ \vec{D} = 7\hat{i} + 9\hat{j} + 11\hat{k} \]