The adjacent sides of a parallelogram are `hat(i) + 2 hat(j) + 3 hat(k) and 2 hat (i) - hat(j) + hat(k)` . Find the unit vectors parallel to diagonals.

Answer

To find the unit vectors parallel to the diagonals of the parallelogram formed by the given adjacent sides, we can follow these steps:

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

### Step 2: Calculate the diagonal vector \( \mathbf{d_1} \)
The first diagonal \( \mathbf{d_1} \) can be found by adding the two vectors:
\[ \mathbf{d_1} = \mathbf{a} + \mathbf{b} \]
Calculating this gives:
\[ \mathbf{d_1} = (\hat{i} + 2\hat{j} + 3\hat{k}) + (2\hat{i} - \hat{j} + \hat{k}) = (1 + 2)\hat{i} + (2 - 1)\hat{j} + (3 + 1)\hat{k} = 3\hat{i} + \hat{j} + 4\hat{k} \]

### Step 3: Find the magnitude of \( \mathbf{d_1} \)
The magnitude of \( \mathbf{d_1} \) is calculated as follows:
\[ |\mathbf{d_1}| = \sqrt{(3)^2 + (1)^2 + (4)^2} = \sqrt{9 + 1 + 16} = \sqrt{26} \]

### Step 4: Calculate the unit vector parallel to diagonal \( \mathbf{d_1} \)
The unit vector \( \mathbf{u_1} \) parallel to diagonal \( \mathbf{d_1} \) is given by:
\[ \mathbf{u_1} = \frac{\mathbf{d_1}}{|\mathbf{d_1}|} = \frac{3\hat{i} + \hat{j} + 4\hat{k}}{\sqrt{26}} = \frac{3}{\sqrt{26}}\hat{i} + \frac{1}{\sqrt{26}}\hat{j} + \frac{4}{\sqrt{26}}\hat{k} \]

### Step 5: Calculate the diagonal vector \( \mathbf{d_2} \)
The second diagonal \( \mathbf{d_2} \) can be found by subtracting the two vectors:
\[ \mathbf{d_2} = \mathbf{b} - \mathbf{a} \]
Calculating this gives:
\[ \mathbf{d_2} = (2\hat{i} - \hat{j} + \hat{k}) - (\hat{i} + 2\hat{j} + 3\hat{k}) = (2 - 1)\hat{i} + (-1 - 2)\hat{j} + (1 - 3)\hat{k} = \hat{i} - 3\hat{j} - 2\hat{k} \]

### Step 6: Find the magnitude of \( \mathbf{d_2} \)
The magnitude of \( \mathbf{d_2} \) is calculated as follows:
\[ |\mathbf{d_2}| = \sqrt{(1)^2 + (-3)^2 + (-2)^2} = \sqrt{1 + 9 + 4} = \sqrt{14} \]

### Step 7: Calculate the unit vector parallel to diagonal \( \mathbf{d_2} \)
The unit vector \( \mathbf{u_2} \) parallel to diagonal \( \mathbf{d_2} \) is given by:
\[ \mathbf{u_2} = \frac{\mathbf{d_2}}{|\mathbf{d_2}|} = \frac{\hat{i} - 3\hat{j} - 2\hat{k}}{\sqrt{14}} = \frac{1}{\sqrt{14}}\hat{i} - \frac{3}{\sqrt{14}}\hat{j} - \frac{2}{\sqrt{14}}\hat{k} \]

### Final Result
The unit vectors parallel to the diagonals of the parallelogram are:
- \( \mathbf{u_1} = \frac{3}{\sqrt{26}}\hat{i} + \frac{1}{\sqrt{26}}\hat{j} + \frac{4}{\sqrt{26}}\hat{k} \)
- \( \mathbf{u_2} = \frac{1}{\sqrt{14}}\hat{i} - \frac{3}{\sqrt{14}}\hat{j} - \frac{2}{\sqrt{14}}\hat{k} \)