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))`

Explanation

To find a unit vector in the direction of the resultant vector of \(\vec{A}\) 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:
\[ \vec{R} = \vec{A} + \vec{B} \]
Substituting the values of \(\vec{A}\) and \(\vec{B}\):
\[ \vec{R} = (-2\hat{i} + 3\hat{j} + \hat{k}) + (\hat{i} + 2\hat{j} - 4\hat{k}) \]
Now, combine the components:
\[ \vec{R} = (-2 + 1)\hat{i} + (3 + 2)\hat{j} + (1 - 4)\hat{k} \]
\[ \vec{R} = -\hat{i} + 5\hat{j} - 3\hat{k} \]

### Step 3: Calculate the magnitude of \(\vec{R}\)
The magnitude of \(\vec{R}\) 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 in the direction of \(\vec{R}\)
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:
\[ \hat{R} = \frac{-\hat{i} + 5\hat{j} - 3\hat{k}}{\sqrt{35}} \]

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