Find a unit vector in the direction of the resultant of the vectors `(hati+2hatj+3hatk),(-hati+2hatj+hatk) and (3hati+hatj)`.

Answer

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

### Step 1: Find the resultant vector
The resultant vector \( \mathbf{R} \) is given by the sum of the vectors \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \).

\[ \mathbf{R} = \mathbf{a} + \mathbf{b} + \mathbf{c} \]

Substituting the values of the vectors:

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

### Step 2: Combine the components
Now, we will combine the \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \) components separately.

- For \( \hat{i} \):
\[ 1 - 1 + 3 = 3\hat{i} \]

- For \( \hat{j} \):
\[ 2 + 2 + 1 = 5\hat{j} \]

- For \( \hat{k} \):
\[ 3 + 1 = 4\hat{k} \]

Thus, the resultant vector is:

\[ \mathbf{R} = 3\hat{i} + 5\hat{j} + 4\hat{k} \]

### Step 3: Calculate the magnitude of the resultant vector
The magnitude \( |\mathbf{R}| \) is calculated using the formula:

\[ |\mathbf{R}| = \sqrt{(3)^2 + (5)^2 + (4)^2} \]

Calculating the squares:

\[ |\mathbf{R}| = \sqrt{9 + 25 + 16} = \sqrt{50} = 5\sqrt{2} \]

### Step 4: Find the unit vector in the direction of the resultant vector
The unit vector \( \hat{r} \) in the direction of \( \mathbf{R} \) is given by:

\[ \hat{r} = \frac{\mathbf{R}}{|\mathbf{R}|} \]

Substituting the values:

\[ \hat{r} = \frac{3\hat{i} + 5\hat{j} + 4\hat{k}}{5\sqrt{2}} \]

This can be written as:

\[ \hat{r} = \frac{3}{5\sqrt{2}} \hat{i} + \frac{5}{5\sqrt{2}} \hat{j} + \frac{4}{5\sqrt{2}} \hat{k} \]

Simplifying further:

\[ \hat{r} = \frac{3}{5\sqrt{2}} \hat{i} + \frac{1}{\sqrt{2}} \hat{j} + \frac{4}{5\sqrt{2}} \hat{k} \]

### Final Result
Thus, the unit vector in the direction of the resultant vector is:

\[ \hat{r} = \frac{3}{5\sqrt{2}} \hat{i} + \frac{1}{\sqrt{2}} \hat{j} + \frac{4}{5\sqrt{2}} \hat{k} \]