Unit vector parallel to the resultant of vectors `A = 4hatj - 3hatj and B =8hatj+8hatj` will be

Options

A. `(24hati+5hatj)/(13)`
B. `(12hati+5hatj)/(13)`
C. `(6hati+5hatj)/(13)`
D. None of these

Explanation

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

### Step 1: Find the resultant vector \( \mathbf{R} \)
The resultant vector \( \mathbf{R} \) is given by the vector sum of \( \mathbf{A} \) and \( \mathbf{B} \).

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

### Step 2: Combine the components
Now, we combine the \( \hat{i} \) and \( \hat{j} \) components separately.

\[ \mathbf{R} = (4 + 8)\hat{i} + (-3 + 8)\hat{j} = 12\hat{i} + 5\hat{j} \]

### Step 3: Calculate the magnitude of the resultant vector \( |\mathbf{R}| \)
The magnitude of the resultant vector \( |\mathbf{R}| \) is calculated using the Pythagorean theorem.

\[ |\mathbf{R}| = \sqrt{(12)^2 + (5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \]

### Step 4: Find the unit vector \( \hat{u} \)
The unit vector \( \hat{u} \) parallel to the resultant vector \( \mathbf{R} \) is given by:

\[ \hat{u} = \frac{\mathbf{R}}{|\mathbf{R}|} \]

Substituting the values we found:

\[ \hat{u} = \frac{12\hat{i} + 5\hat{j}}{13} \]

### Final Result
Thus, the unit vector parallel to the resultant of vectors \( \mathbf{A} \) and \( \mathbf{B} \) is:

\[ \hat{u} = \frac{12}{13}\hat{i} + \frac{5}{13}\hat{j} \]