Unit vector parallel to the resultant of vector `8hati` and `8hatj` will be

Options

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

Explanation

To find the unit vector parallel to the resultant of the vectors \(8\hat{i}\) and \(8\hat{j}\), we can follow these steps:

### Step 1: Identify the vectors
Let vector \( \mathbf{A} = 8\hat{i} \) and vector \( \mathbf{B} = 8\hat{j} \).

### Step 2: Calculate the resultant vector
The resultant vector \( \mathbf{R} \) is given by the sum of the two vectors:
\[ \mathbf{R} = \mathbf{A} + \mathbf{B} = 8\hat{i} + 8\hat{j} \]

### Step 3: Find the magnitude of the resultant vector
The magnitude of the resultant vector \( \mathbf{R} \) can be calculated using the formula:
\[ |\mathbf{R}| = \sqrt{(R_x)^2 + (R_y)^2} \]
where \( R_x \) and \( R_y \) are the components of the resultant vector. Here, \( R_x = 8 \) and \( R_y = 8 \):
\[ |\mathbf{R}| = \sqrt{(8)^2 + (8)^2} = \sqrt{64 + 64} = \sqrt{128} = 8\sqrt{2} \]

### Step 4: Calculate the unit vector
The unit vector \( \hat{u} \) in the direction of the resultant vector is given by:
\[ \hat{u} = \frac{\mathbf{R}}{|\mathbf{R}|} \]
Substituting the values we found:
\[ \hat{u} = \frac{8\hat{i} + 8\hat{j}}{8\sqrt{2}} = \frac{8}{8\sqrt{2}} \hat{i} + \frac{8}{8\sqrt{2}} \hat{j} = \frac{1}{\sqrt{2}} \hat{i} + \frac{1}{\sqrt{2}} \hat{j} \]
This can be simplified to:
\[ \hat{u} = \hat{i} + \hat{j} \quad \text{(divided by } \sqrt{2} \text{)} \]

### Final Answer
Thus, the unit vector parallel to the resultant of the vectors \(8\hat{i}\) and \(8\hat{j}\) is:
\[ \hat{u} = \frac{\hat{i} + \hat{j}}{\sqrt{2}} \]

Since this does not match any of the provided options, the correct answer is "none of these".

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