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

Options

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

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: Calculate the first diagonal vector \( \mathbf{d_1} \)
The first diagonal of the parallelogram can be found by adding the two vectors \( \mathbf{a} \) and \( \mathbf{b} \):

\[ \mathbf{d_1} = \mathbf{a} + \mathbf{b} \]

Substituting the values of \( \mathbf{a} \) and \( \mathbf{b} \):

\[ \mathbf{d_1} = (2\hat{i} + 4\hat{j} - 5\hat{k}) + (\hat{i} + 2\hat{j} + 3\hat{k}) \]

Now, combine the components:

\[ \mathbf{d_1} = (2 + 1)\hat{i} + (4 + 2)\hat{j} + (-5 + 3)\hat{k} = 3\hat{i} + 6\hat{j} - 2\hat{k} \]

### Step 2: Calculate the magnitude of \( \mathbf{d_1} \)
The magnitude of the diagonal vector \( \mathbf{d_1} \) is given by:

\[ |\mathbf{d_1}| = \sqrt{(3)^2 + (6)^2 + (-2)^2} \]

Calculating each component:

\[ |\mathbf{d_1}| = \sqrt{9 + 36 + 4} = \sqrt{49} = 7 \]

### Step 3: Calculate the unit vector along \( \mathbf{d_1} \)
The unit vector \( \mathbf{d_1}^{\hat{}} \) in the direction of \( \mathbf{d_1} \) is given by:

\[ \mathbf{d_1}^{\hat{}} = \frac{\mathbf{d_1}}{|\mathbf{d_1}|} \]

Substituting the values:

\[ \mathbf{d_1}^{\hat{}} = \frac{3\hat{i} + 6\hat{j} - 2\hat{k}}{7} = \frac{3}{7}\hat{i} + \frac{6}{7}\hat{j} - \frac{2}{7}\hat{k} \]

### Final Result
The unit vector parallel to one of the diagonals of the parallelogram is:

\[ \mathbf{d_1}^{\hat{}} = \frac{3}{7}\hat{i} + \frac{6}{7}\hat{j} - \frac{2}{7}\hat{k} \]

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