A unit vector in the direction of resultant vector of `vecA = -2hati + 3hatj + hatk and vecB = hati + 2 hatj - 4 hatk` is

Options

A. `(-2hati+3hatj+hatk)/(sqrt(35))`
B. `(hati+2hatj-4hatk)/(sqrt(35))`
C. `(-hati+5hatj-3hatk)/(sqrt(35))`
D. `(-3hati+hatj+5hatk)/(sqrt(35))`

Explanation

To find the unit vector in the direction of the resultant vector of the vectors \(\vec{A}\) and \(\vec{B}\), we will follow these steps:

### Step 1: Write down the vectors
Given:
\[ \vec{A} = -2\hat{i} + 3\hat{j} + \hat{k} \]
\[ \vec{B} = \hat{i} + 2\hat{j} - 4\hat{k} \]

### Step 2: Calculate the resultant vector \(\vec{R}\)
The resultant vector \(\vec{R}\) is given by the sum of \(\vec{A}\) and \(\vec{B}\):
\[ \vec{R} = \vec{A} + \vec{B} \]
Calculating the components:
- For the \(\hat{i}\) component:
\[ -2 + 1 = -1 \]
- For the \(\hat{j}\) component:
\[ 3 + 2 = 5 \]
- For the \(\hat{k}\) component:
\[ 1 - 4 = -3 \]
Thus, the resultant vector is:
\[ \vec{R} = -\hat{i} + 5\hat{j} - 3\hat{k} \]

### Step 3: Find the magnitude of the resultant vector \(\vec{R}\)
The magnitude \(|\vec{R}|\) is calculated using the formula:
\[ |\vec{R}| = \sqrt{(-1)^2 + (5)^2 + (-3)^2} \]
Calculating each term:
- \((-1)^2 = 1\)
- \(5^2 = 25\)
- \((-3)^2 = 9\)

Adding these values:
\[ |\vec{R}| = \sqrt{1 + 25 + 9} = \sqrt{35} \]

### Step 4: Calculate the unit vector \(\hat{R}\)
The unit vector \(\hat{R}\) in the direction of the resultant vector \(\vec{R}\) is given by:
\[ \hat{R} = \frac{\vec{R}}{|\vec{R}|} \]
Substituting the values:
\[ \hat{R} = \frac{-\hat{i} + 5\hat{j} - 3\hat{k}}{\sqrt{35}} \]

### Step 5: Final expression for the unit vector
Thus, the unit vector in the direction of the resultant vector is:
\[ \hat{R} = \frac{-1}{\sqrt{35}}\hat{i} + \frac{5}{\sqrt{35}}\hat{j} - \frac{3}{\sqrt{35}}\hat{k} \]

### Conclusion
The unit vector in the direction of the resultant vector of \(\vec{A}\) and \(\vec{B}\) is:
\[ \hat{R} = \frac{-1}{\sqrt{35}}\hat{i} + \frac{5}{\sqrt{35}}\hat{j} - \frac{3}{\sqrt{35}}\hat{k} \]