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

Options

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

Explanation

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

### Step 1: Identify the vector components
The vector \(\vec{A}\) has components:
- \(A_x = 2\) (the coefficient of \(\hat{i}\))
- \(A_y = 3\) (the coefficient of \(\hat{j}\))

### Step 2: Calculate the magnitude of the vector \(\vec{A}\)
The magnitude of a vector \(\vec{A}\) is given by the formula:
\[ |\vec{A}| = \sqrt{A_x^2 + A_y^2} \]
Substituting the values:
\[ |\vec{A}| = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \]

### Step 3: Calculate the unit vector \(\hat{A}\)
The unit vector \(\hat{A}\) in the direction of \(\vec{A}\) is given by:
\[ \hat{A} = \frac{\vec{A}}{|\vec{A}|} \]
Substituting \(\vec{A}\) and its magnitude:
\[ \hat{A} = \frac{2\hat{i} + 3\hat{j}}{\sqrt{13}} \]

### 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 \(\vec{A}\) is:
\[ \hat{A} = \frac{2}{\sqrt{13}} \hat{i} + \frac{3}{\sqrt{13}} \hat{j} \]

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