The unit vector along `vec(A)= 2 hat i + 3 hat j` is :

Options

A. `2 hat i+ 3 hat j`
B. `(2 hati +3 hatj)/(2)`
C. `(2 hati +3 hatj)/(3)`
D. `(2 hati+3hatj)/(sqrt(13))`

Explanation

To find the unit vector along the vector \(\vec{A} = 2 \hat{i} + 3 \hat{j}\), we follow these steps:

### Step 1: Identify the vector
The given vector is:
\[ \vec{A} = 2 \hat{i} + 3 \hat{j} \]

### 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:
\[ |\vec{A}| = \sqrt{a^2 + b^2} \]
In our case, \(a = 2\) and \(b = 3\). Thus, we calculate:
\[ |\vec{A}| = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \]

### Step 3: Find 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:
\[ \hat{A} = \frac{2 \hat{i} + 3 \hat{j}}{\sqrt{13}} \]

### Step 4: Write the final expression for the unit vector
Thus, the unit vector along \(\vec{A}\) is:
\[ \hat{A} = \frac{2}{\sqrt{13}} \hat{i} + \frac{3}{\sqrt{13}} \hat{j} \]

### Final Answer
The unit vector along \(\vec{A} = 2 \hat{i} + 3 \hat{j}\) is:
\[ \hat{A} = \frac{2}{\sqrt{13}} \hat{i} + \frac{3}{\sqrt{13}} \hat{j} \]

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