The unit vector 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 will follow these steps:

### Step 1: Calculate the Resultant Vector
The resultant vector \(\vec{R}\) is given by the vector sum of \(\vec{A}\) and \(\vec{B}\).

\[ \vec{R} = \vec{A} + \vec{B} \]

Substituting the values of \(\vec{A}\) and \(\vec{B}\):

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

### Step 2: Combine Like Terms
Now, we will combine the components of \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\):

\[ \vec{R} = (4 - 1)\hat{i} + (3 + 3)\hat{j} + (6 - 8)\hat{k} \]

Calculating each component:

\[ \vec{R} = 3\hat{i} + 6\hat{j} - 2\hat{k} \]

### Step 3: Calculate the Magnitude of the Resultant Vector
The magnitude of \(\vec{R}\) is given by:

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

Calculating the squares:

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

### 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:

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

### Step 5: Write the Final Unit Vector
Thus, the unit vector parallel to the resultant vector is:

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

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

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

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