The sides of a parallelogram are `2hati +4hatj -5hatk and hati + 2hatj +3hatk `. The unit vector parallel to one of the diagonals is

Options

A. `(1)/(7)(3hati +6hatj-2hatk)`
B. `(1)/(7)(3hati-6hatj -2hatk)`
C. `(1)/(sqrt(69))(hati +2hatj +8hatk)`
D. `(1)/(sqrt(69))(-hati -2hatj +8hatk)`

Explanation

To find the unit vector parallel to one of the diagonals of the parallelogram defined by the vectors \( \mathbf{a} = 2\hat{i} + 4\hat{j} - 5\hat{k} \) and \( \mathbf{b} = \hat{i} + 2\hat{j} + 3\hat{k} \), we will follow these steps:

### Step 1: Identify the vectors
Let:
- \( \mathbf{a} = 2\hat{i} + 4\hat{j} - 5\hat{k} \)
- \( \mathbf{b} = \hat{i} + 2\hat{j} + 3\hat{k} \)

### Step 2: Calculate the diagonals
The diagonals of the parallelogram can be calculated as:
1. \( \mathbf{p} = \mathbf{a} + \mathbf{b} \)
2. \( \mathbf{q} = \mathbf{b} - \mathbf{a} \)

Calculating \( \mathbf{p} \):
\[ \mathbf{p} = (2\hat{i} + 4\hat{j} - 5\hat{k}) + (\hat{i} + 2\hat{j} + 3\hat{k}) \]
\[ = (2 + 1)\hat{i} + (4 + 2)\hat{j} + (-5 + 3)\hat{k} \]
\[ = 3\hat{i} + 6\hat{j} - 2\hat{k} \]

Calculating \( \mathbf{q} \):
\[ \mathbf{q} = (\hat{i} + 2\hat{j} + 3\hat{k}) - (2\hat{i} + 4\hat{j} - 5\hat{k}) \]
\[ = (1 - 2)\hat{i} + (2 - 4)\hat{j} + (3 + 5)\hat{k} \]
\[ = -\hat{i} - 2\hat{j} + 8\hat{k} \]

### Step 3: Find the magnitudes of the diagonals
Now we will find the magnitudes of \( \mathbf{p} \) and \( \mathbf{q} \).

Magnitude of \( \mathbf{p} \):
\[ |\mathbf{p}| = \sqrt{(3)^2 + (6)^2 + (-2)^2} = \sqrt{9 + 36 + 4} = \sqrt{49} = 7 \]

Magnitude of \( \mathbf{q} \):
\[ |\mathbf{q}| = \sqrt{(-1)^2 + (-2)^2 + (8)^2} = \sqrt{1 + 4 + 64} = \sqrt{69} \]

### Step 4: Calculate the unit vectors
The unit vector parallel to diagonal \( \mathbf{p} \):
\[ \hat{u}_p = \frac{\mathbf{p}}{|\mathbf{p}|} = \frac{3\hat{i} + 6\hat{j} - 2\hat{k}}{7} = \frac{3}{7}\hat{i} + \frac{6}{7}\hat{j} - \frac{2}{7}\hat{k} \]

The unit vector parallel to diagonal \( \mathbf{q} \):
\[ \hat{u}_q = \frac{\mathbf{q}}{|\mathbf{q}|} = \frac{-\hat{i} - 2\hat{j} + 8\hat{k}}{\sqrt{69}} = -\frac{1}{\sqrt{69}}\hat{i} - \frac{2}{\sqrt{69}}\hat{j} + \frac{8}{\sqrt{69}}\hat{k} \]

### Final Answer
The unit vectors parallel to the diagonals of the parallelogram are:
1. \( \hat{u}_p = \frac{3}{7}\hat{i} + \frac{6}{7}\hat{j} - \frac{2}{7}\hat{k} \)
2. \( \hat{u}_q = -\frac{1}{\sqrt{69}}\hat{i} - \frac{2}{\sqrt{69}}\hat{j} + \frac{8}{\sqrt{69}}\hat{k} \)