If the sides AB and AD of a parallelogram ABCD are represented by the vector `2 hat(i) + 4 hat(j) - 5 hat(k)` and `hat(i) + 2 hat(j) + 3 hat(k)` , then a unit vector along `vec( AC )` is

Options

A. `(1)/( 3) ( 3 hat(i) - 6 hat(j) - 2 hat(k))`
B. `(1)/(3) ( 3 hat(i) + 6 hat(j) - 2 hat(k))`
C. `(1)/( 7) ( 3 hat(i) - 6 hat(j) - 2 hat(k))`
D. `(1)/( 7) ( 3 hat(i) + 6 hat(j) - 2 hat(k))`

Answer

To find a unit vector along the diagonal \( \vec{AC} \) of the parallelogram ABCD, we will follow these steps:

### Step 1: Identify the vectors representing the sides of the parallelogram
The sides of the parallelogram are given as:
- \( \vec{AB} = 2\hat{i} + 4\hat{j} - 5\hat{k} \)
- \( \vec{AD} = \hat{i} + 2\hat{j} + 3\hat{k} \)

### Step 2: Find the vector \( \vec{AC} \)
In a parallelogram, the diagonal \( \vec{AC} \) can be found by adding the vectors \( \vec{AB} \) and \( \vec{AD} \):
\[ \vec{AC} = \vec{AB} + \vec{AD} \]
Substituting the values:
\[ \vec{AC} = (2\hat{i} + 4\hat{j} - 5\hat{k}) + (\hat{i} + 2\hat{j} + 3\hat{k}) \]
Now, combine the like terms:
\[ \vec{AC} = (2 + 1)\hat{i} + (4 + 2)\hat{j} + (-5 + 3)\hat{k} \]
\[ \vec{AC} = 3\hat{i} + 6\hat{j} - 2\hat{k} \]

### Step 3: Calculate the magnitude of \( \vec{AC} \)
The magnitude of a vector \( \vec{v} = a\hat{i} + b\hat{j} + c\hat{k} \) is given by:
\[ |\vec{v}| = \sqrt{a^2 + b^2 + c^2} \]
For \( \vec{AC} = 3\hat{i} + 6\hat{j} - 2\hat{k} \):
\[ |\vec{AC}| = \sqrt{3^2 + 6^2 + (-2)^2} \]
Calculating each term:
\[ |\vec{AC}| = \sqrt{9 + 36 + 4} = \sqrt{49} = 7 \]

### Step 4: Find the unit vector along \( \vec{AC} \)
The unit vector \( \hat{u} \) in the direction of \( \vec{AC} \) is given by:
\[ \hat{u} = \frac{\vec{AC}}{|\vec{AC}|} \]
Substituting the values we found:
\[ \hat{u} = \frac{3\hat{i} + 6\hat{j} - 2\hat{k}}{7} \]
Thus, the unit vector along \( \vec{AC} \) is:
\[ \hat{u} = \frac{3}{7}\hat{i} + \frac{6}{7}\hat{j} - \frac{2}{7}\hat{k} \]

### Final Answer
The unit vector along \( \vec{AC} \) is:
\[ \hat{u} = \frac{3}{7}\hat{i} + \frac{6}{7}\hat{j} - \frac{2}{7}\hat{k} \]