The component of vector `A=2hati+3hatj` along the vector `hati+hatj` is

Options

A. `(5)/(sqrt(2))`
B. `4sqrt(2)`
C. `(sqrt(2))/(3)`
D. None of these

Explanation

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

### Step 1: Calculate the Magnitude of Vector A
The magnitude of vector \( A \) is calculated using the formula:
\[ |A| = \sqrt{(A_x)^2 + (A_y)^2} \]
where \( A_x \) and \( A_y \) are the components of vector \( A \).

For \( A = 2\hat{i} + 3\hat{j} \):
\[ |A| = \sqrt{(2)^2 + (3)^2} = \sqrt{4 + 9} = \sqrt{13} \]

### Step 2: Calculate the Magnitude of Vector B
Similarly, we calculate the magnitude of vector \( B \):
\[ |B| = \sqrt{(B_x)^2 + (B_y)^2} \]
where \( B_x \) and \( B_y \) are the components of vector \( B \).

For \( B = \hat{i} + \hat{j} \):
\[ |B| = \sqrt{(1)^2 + (1)^2} = \sqrt{1 + 1} = \sqrt{2} \]

### Step 3: Calculate the Dot Product of A and B
The dot product \( A \cdot B \) is given by:
\[ A \cdot B = A_x B_x + A_y B_y \]

For \( A = 2\hat{i} + 3\hat{j} \) and \( B = \hat{i} + \hat{j} \):
\[ A \cdot B = (2)(1) + (3)(1) = 2 + 3 = 5 \]

### Step 4: Calculate the Component of A along B
The component of vector \( A \) along vector \( B \) is given by:
\[ \text{Component of } A \text{ along } B = \frac{A \cdot B}{|B|} \]

Substituting the values we calculated:
\[ \text{Component of } A \text{ along } B = \frac{5}{\sqrt{2}} \]

### Step 5: Final Calculation
To express the result in a more standard form, we can rationalize the denominator:
\[ \text{Component of } A \text{ along } B = \frac{5}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{5\sqrt{2}}{2} \]

### Conclusion
The component of vector \( A \) along vector \( B \) is \( \frac{5}{\sqrt{2}} \).