A unit vector in the dirction of resultant vector of `vec(A)= -2hat(i)+3hat(j)+hat(k)` and `vec(B)= hat(i)+2hat(j)-4hat(k)` is

Options

A. `(-2hat(i) + 3hat(j) + hat(k))/(sqrt(35))`
B. `(-hat(i) + 2hat(j) + 4hat(k))/(sqrt(35))`
C. `(-hat(i) + 5hat(j) - 3hat(k))/(sqrt(35))`
D. `(-3hat(i) + hat(j) - 5hat(k))/(sqrt(35))`

Answer

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

### Step 1: Find the Resultant Vector
The resultant vector \(\vec{R}\) is given by the sum of vectors \(\vec{A}\) and \(\vec{B}\):
\[ \vec{R} = \vec{A} + \vec{B} \]
Substituting the values of \(\vec{A}\) and \(\vec{B}\):
\[ \vec{R} = (-2\hat{i} + 3\hat{j} + \hat{k}) + (\hat{i} + 2\hat{j} - 4\hat{k}) \]

### Step 2: Combine the Components
Now, we combine the components of \(\vec{A}\) and \(\vec{B}\):
- 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: Calculate the Magnitude of the Resultant Vector
The magnitude of the resultant vector \(\vec{R}\) is given by:
\[ |\vec{R}| = \sqrt{(-1)^2 + (5)^2 + (-3)^2} \]
Calculating each term:
\[ |\vec{R}| = \sqrt{1 + 25 + 9} = \sqrt{35} \]

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

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