Fina a unit vector in the direaction of the resultant of the vectors `hati - hatj + 3hatk, 2 hati + hatj- 2 hatk and hati + 2 hatj- 2 hatk.`

Answer

To find a unit vector in the direction of the resultant of the given vectors \( \hat{i} - \hat{j} + 3\hat{k} \), \( 2\hat{i} + \hat{j} - 2\hat{k} \), and \( \hat{i} + 2\hat{j} - 2\hat{k} \), we will follow these steps:

### Step 1: Write down the vectors
Let:
- \( \vec{A} = \hat{i} - \hat{j} + 3\hat{k} \)
- \( \vec{B} = 2\hat{i} + \hat{j} - 2\hat{k} \)
- \( \vec{C} = \hat{i} + 2\hat{j} - 2\hat{k} \)

### Step 2: Find the resultant vector \( \vec{R} \)
The resultant vector \( \vec{R} \) is the sum of the three vectors:
\[ \vec{R} = \vec{A} + \vec{B} + \vec{C} \]
Calculating the components:
- For the \( \hat{i} \) component: \( 1 + 2 + 1 = 4 \)
- For the \( \hat{j} \) component: \( -1 + 1 + 2 = 2 \)
- For the \( \hat{k} \) component: \( 3 - 2 - 2 = -1 \)

Thus, the resultant vector is:
\[ \vec{R} = 4\hat{i} + 2\hat{j} - 1\hat{k} \]

### Step 3: Calculate the magnitude of the resultant vector \( |\vec{R}| \)
The magnitude of \( \vec{R} \) is given by:
\[ |\vec{R}| = \sqrt{(4)^2 + (2)^2 + (-1)^2} \]
Calculating:
\[ |\vec{R}| = \sqrt{16 + 4 + 1} = \sqrt{21} \]

### Step 4: Find the unit vector in the direction of \( \vec{R} \)
The unit vector \( \hat{u} \) in the direction of \( \vec{R} \) is given by:
\[ \hat{u} = \frac{\vec{R}}{|\vec{R}|} \]
Substituting the values:
\[ \hat{u} = \frac{4\hat{i} + 2\hat{j} - 1\hat{k}}{\sqrt{21}} \]

### Final Result
Thus, the unit vector in the direction of the resultant vector is:
\[ \hat{u} = \frac{4}{\sqrt{21}} \hat{i} + \frac{2}{\sqrt{21}} \hat{j} - \frac{1}{\sqrt{21}} \hat{k} \]