The unit vactor parallel to the resultant of the vectors `vec(A)=4hat(i)+3hat(j)+6hat(k)` and `vec(B)=-hat(i)+3hat(j)-8hat(k)` is :-

Options

A. `1/7 (3hat(i)+6hat(j)-2hat(k))`
B. `1/7 (3hat(i)+6hat(j)+2hat(k))`
C. `1/49 (3hat(i)+6hat(j)+2hat(k))`
D. `1/49(3hat(i)+6hat(j)-2hat(k))`

Explanation

To find the unit vector parallel to the resultant of the vectors \(\vec{A} = 4\hat{i} + 3\hat{j} + 6\hat{k}\) and \(\vec{B} = -\hat{i} + 3\hat{j} - 8\hat{k}\), we can follow these steps:

### Step 1: Find the Resultant Vector
To find the resultant vector \(\vec{R}\), we add the components of \(\vec{A}\) and \(\vec{B}\):

\[ \vec{R} = \vec{A} + \vec{B} = (4\hat{i} + 3\hat{j} + 6\hat{k}) + (-\hat{i} + 3\hat{j} - 8\hat{k}) \]

Calculating the components:

- For the \(\hat{i}\) component:
\[ 4 - 1 = 3 \]

- For the \(\hat{j}\) component:
\[ 3 + 3 = 6 \]

- For the \(\hat{k}\) component:
\[ 6 - 8 = -2 \]

Thus, the resultant vector is:
\[ \vec{R} = 3\hat{i} + 6\hat{j} - 2\hat{k} \]

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

\[ |\vec{R}| = \sqrt{(3)^2 + (6)^2 + (-2)^2} \]

Calculating each term:

\[ = \sqrt{9 + 36 + 4} = \sqrt{49} = 7 \]

### Step 3: 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 the values we found:

\[ \hat{u} = \frac{3\hat{i} + 6\hat{j} - 2\hat{k}}{7} \]

This simplifies to:

\[ \hat{u} = \frac{3}{7}\hat{i} + \frac{6}{7}\hat{j} - \frac{2}{7}\hat{k} \]

### Final Answer
The unit vector parallel to the resultant of the vectors \(\vec{A}\) and \(\vec{B}\) is:

\[ \hat{u} = \frac{3}{7}\hat{i} + \frac{6}{7}\hat{j} - \frac{2}{7}\hat{k} \]