A unit vector parallel to the resultant of the vectors
`overset(harr)A=4overset^j + 3 overset ^j + 6 overset ^k" and" overset(harr)B=- overset ^i + 8 overset ^j-8 overset ^k`

Options

A. `(3 overset ^i + 11 overset ^j -2 overset ^k)/(2)`
B. `( overset ^ i + 2 overset ^ j- 3 overset ^k)/(sqrt 166)`
C. `(3 overset ^ i + 11 overset ^j-2 overset ^k)/(sqrt (134))`
D. `(4 overset ^i + 6 overset ^j + 8 overset ^k)/(sqrt 11)`

Explanation

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

### Step 1: Write down the vectors
Given:
\[ \vec{A} = 4\hat{i} + 3\hat{j} + 6\hat{k} \]
\[ \vec{B} = -\hat{i} + 8\hat{j} - 8\hat{k} \]

### Step 2: Find the resultant vector \( \vec{R} \)
The resultant vector \( \vec{R} \) is obtained by adding \( \vec{A} \) and \( \vec{B} \):
\[ \vec{R} = \vec{A} + \vec{B} = (4\hat{i} + 3\hat{j} + 6\hat{k}) + (-\hat{i} + 8\hat{j} - 8\hat{k}) \]

### Step 3: Combine like terms
Combine the coefficients of \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \):
\[ \vec{R} = (4 - 1)\hat{i} + (3 + 8)\hat{j} + (6 - 8)\hat{k} \]
\[ \vec{R} = 3\hat{i} + 11\hat{j} - 2\hat{k} \]

### Step 4: Calculate the magnitude of the resultant vector \( \vec{R} \)
The magnitude \( |\vec{R}| \) is given by:
\[ |\vec{R}| = \sqrt{(3)^2 + (11)^2 + (-2)^2} \]
\[ |\vec{R}| = \sqrt{9 + 121 + 4} = \sqrt{134} \]

### Step 5: Find the unit vector
The unit vector \( \hat{u} \) in the direction of \( \vec{R} \) is given by:
\[ \hat{u} = \frac{\vec{R}}{|\vec{R}|} \]
Substituting \( \vec{R} \) and \( |\vec{R}| \):
\[ \hat{u} = \frac{3\hat{i} + 11\hat{j} - 2\hat{k}}{\sqrt{134}} \]

### Final Answer
Thus, the unit vector parallel to the resultant of the vectors \( \vec{A} \) and \( \vec{B} \) is:
\[ \hat{u} = \frac{3}{\sqrt{134}}\hat{i} + \frac{11}{\sqrt{134}}\hat{j} - \frac{2}{\sqrt{134}}\hat{k} \]