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

### Step 1: Find the Resultant Vector
The resultant vector \(\vec{C}\) is given by the sum of vectors \(\vec{A}\) and \(\vec{B}\):
\[ \vec{C} = \vec{A} + \vec{B} \]
Substituting the values:
\[ \vec{C} = (4\hat{i} - 3\hat{j}) + (8\hat{i} + 8\hat{j}) \]
Combine the components:
\[ \vec{C} = (4 + 8)\hat{i} + (-3 + 8)\hat{j} = 12\hat{i} + 5\hat{j} \]

### Step 2: Calculate the Magnitude of the Resultant Vector
The magnitude of vector \(\vec{C}\) is calculated using the formula:
\[ |\vec{C}| = \sqrt{C_x^2 + C_y^2} \]
Where \(C_x\) and \(C_y\) are the components of \(\vec{C}\):
\[ |\vec{C}| = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \]

### Step 3: Find the Unit Vector
The unit vector \(\hat{C}\) in the direction of \(\vec{C}\) is given by:
\[ \hat{C} = \frac{\vec{C}}{|\vec{C}|} \]
Substituting the values we have:
\[ \hat{C} = \frac{12\hat{i} + 5\hat{j}}{13} \]

### Final Answer
Thus, the unit vector parallel to the resultant of vectors \(\vec{A}\) and \(\vec{B}\) is:
\[ \hat{C} = \frac{12}{13}\hat{i} + \frac{5}{13}\hat{j} \]