Unit vector along `3hat(i)+3hat(j)` is

Options

A. `(hat(i)+hat(j))/(sqrt(2))`
B. `(3hat(i)+3hat(j))/(2)`
C. `hat(i)+hat(j)`
D. `(hat(i)+hat(j))/(sqrt(3))`

Explanation

To find the unit vector along the vector \( \vec{A} = 3\hat{i} + 3\hat{j} \), we will follow these steps:

### Step 1: Identify the vector
We have the vector:
\[ \vec{A} = 3\hat{i} + 3\hat{j} \]

### Step 2: Calculate the magnitude of the vector
The magnitude of a vector \( \vec{A} = a\hat{i} + b\hat{j} \) is given by the formula:
\[ |\vec{A}| = \sqrt{a^2 + b^2} \]
For our vector:
- \( a = 3 \)
- \( b = 3 \)

Now, substituting these values:
\[ |\vec{A}| = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} \]
This can be simplified as:
\[ |\vec{A}| = \sqrt{9 \times 2} = 3\sqrt{2} \]

### Step 3: Calculate the unit vector
The unit vector \( \hat{A} \) in the direction of \( \vec{A} \) is given by:
\[ \hat{A} = \frac{\vec{A}}{|\vec{A}|} \]
Substituting the values we have:
\[ \hat{A} = \frac{3\hat{i} + 3\hat{j}}{3\sqrt{2}} \]

### Step 4: Simplify the unit vector
We can simplify this expression:
\[ \hat{A} = \frac{3}{3\sqrt{2}} \hat{i} + \frac{3}{3\sqrt{2}} \hat{j} = \frac{1}{\sqrt{2}} \hat{i} + \frac{1}{\sqrt{2}} \hat{j} \]
This can also be written as:
\[ \hat{A} = \hat{i} + \hat{j} \text{ divided by } \sqrt{2} \]

### Final Result
Thus, the unit vector along \( 3\hat{i} + 3\hat{j} \) is:
\[ \hat{A} = \frac{1}{\sqrt{2}} \hat{i} + \frac{1}{\sqrt{2}} \hat{j} \]