If `a=hati+2hatj+3hatk,b=-hati+2hatj+hatk and c=3hati+hatj`, then the unit vector along its resultant is

Options

A. `3hati+5hatj+4hatk`
B. `(3hati+5hatj+4hatk)/(50)`
C. `(3hati+5hatj+4hatk)/(5sqrt(2))`
D. none of these

Explanation

To find the unit vector along the resultant of the vectors \( \mathbf{a}, \mathbf{b}, \) and \( \mathbf{c} \), we will follow these steps:

### Step 1: Write down the vectors
Given:
\[ \mathbf{a} = \hat{i} + 2\hat{j} + 3\hat{k} \]
\[ \mathbf{b} = -\hat{i} + 2\hat{j} + \hat{k} \]
\[ \mathbf{c} = 3\hat{i} + \hat{j} \]

### Step 2: Calculate the resultant vector \( \mathbf{R} \)
The resultant vector \( \mathbf{R} \) is given by:
\[ \mathbf{R} = \mathbf{a} + \mathbf{b} + \mathbf{c} \]

Calculating the components:
- **i-component**:
\[ 1 - 1 + 3 = 3 \]
- **j-component**:
\[ 2 + 2 + 1 = 5 \]
- **k-component**:
\[ 3 + 1 + 0 = 4 \]

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 \( |\mathbf{R}| \)
The magnitude \( |\mathbf{R}| \) is calculated as:
\[ |\mathbf{R}| = \sqrt{(3)^2 + (5)^2 + (4)^2} \]
Calculating each term:
\[ |\mathbf{R}| = \sqrt{9 + 25 + 16} = \sqrt{50} \]

### Step 4: Find the unit vector along \( \mathbf{R} \)
The unit vector \( \hat{r} \) along \( \mathbf{R} \) is given by:
\[ \hat{r} = \frac{\mathbf{R}}{|\mathbf{R}|} \]
Substituting the values:
\[ \hat{r} = \frac{3\hat{i} + 5\hat{j} + 4\hat{k}}{\sqrt{50}} = \frac{3\hat{i} + 5\hat{j} + 4\hat{k}}{5\sqrt{2}} \]

### Final Answer
Thus, the unit vector along the resultant is:
\[ \hat{r} = \frac{3}{5\sqrt{2}} \hat{i} + \frac{5}{5\sqrt{2}} \hat{j} + \frac{4}{5\sqrt{2}} \hat{k} \]

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