The unit vector parallel to the resultant of the vectors `2hati+3hatj-hatk` and `4hati-3hatj+2hatk` is

Options

A. `1/sqrt(37)(6hati+hatk)`
B. `1/sqrt(37)(6hati+hatj)`
C. `1/sqrt(37)(6hati+hatk)`
D. none of these

Explanation

To find the unit vector parallel to the resultant of the vectors \( \vec{A} = 2\hat{i} + 3\hat{j} - \hat{k} \) and \( \vec{B} = 4\hat{i} - 3\hat{j} + 2\hat{k} \), we will follow these steps:

### Step 1: Find the resultant vector \( \vec{R} \)
The resultant vector \( \vec{R} \) is obtained by adding the corresponding components of vectors \( \vec{A} \) and \( \vec{B} \).

\[ \vec{R} = \vec{A} + \vec{B} = (2\hat{i} + 3\hat{j} - \hat{k}) + (4\hat{i} - 3\hat{j} + 2\hat{k}) \]

Calculating the components:

- For \( \hat{i} \): \( 2 + 4 = 6 \)
- For \( \hat{j} \): \( 3 - 3 = 0 \)
- For \( \hat{k} \): \( -1 + 2 = 1 \)

Thus, the resultant vector is:

\[ \vec{R} = 6\hat{i} + 0\hat{j} + 1\hat{k} \]

### Step 2: Calculate the magnitude of the resultant vector \( |\vec{R}| \)
The magnitude of \( \vec{R} \) is given by:

\[ |\vec{R}| = \sqrt{(6)^2 + (0)^2 + (1)^2} = \sqrt{36 + 0 + 1} = \sqrt{37} \]

### Step 3: Find the unit vector \( \hat{r} \) in the direction of \( \vec{R} \)
The unit vector \( \hat{r} \) is calculated by dividing the resultant vector \( \vec{R} \) by its magnitude \( |\vec{R}| \):

\[ \hat{r} = \frac{\vec{R}}{|\vec{R}|} = \frac{6\hat{i} + 0\hat{j} + 1\hat{k}}{\sqrt{37}} \]

This simplifies to:

\[ \hat{r} = \frac{6}{\sqrt{37}}\hat{i} + 0\hat{j} + \frac{1}{\sqrt{37}}\hat{k} \]

### Final Answer
The unit vector parallel to the resultant of the given vectors is:

\[ \hat{r} = \frac{6}{\sqrt{37}}\hat{i} + 0\hat{j} + \frac{1}{\sqrt{37}}\hat{k} \]

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