The component of vector `A= 2hat(i)+3hat(j)` along the vector `hat(i)+hat(j)` is

Options

A. `5/(sqrt(2))`
B. `10sqrt(2)`
C. `5sqrt(2)`
D. 5

Explanation

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

### Step 1: Identify the vectors
We have:
- \( \mathbf{A} = 2\hat{i} + 3\hat{j} \)
- \( \mathbf{B} = \hat{i} + \hat{j} \)

### Step 2: Calculate the unit vector of \( \mathbf{B} \)
The unit vector \( \hat{B} \) is given by:
\[ \hat{B} = \frac{\mathbf{B}}{|\mathbf{B}|} \]
First, we need to find the magnitude of \( \mathbf{B} \):
\[ |\mathbf{B}| = \sqrt{(1^2 + 1^2)} = \sqrt{2} \]
Now, we can find the unit vector:
\[ \hat{B} = \frac{\hat{i} + \hat{j}}{\sqrt{2}} = \frac{1}{\sqrt{2}} \hat{i} + \frac{1}{\sqrt{2}} \hat{j} \]

### Step 3: Calculate the dot product \( \mathbf{A} \cdot \hat{B} \)
The dot product is calculated as follows:
\[ \mathbf{A} \cdot \hat{B} = (2\hat{i} + 3\hat{j}) \cdot \left(\frac{1}{\sqrt{2}} \hat{i} + \frac{1}{\sqrt{2}} \hat{j}\right) \]
Calculating the dot product:
\[ \mathbf{A} \cdot \hat{B} = 2 \cdot \frac{1}{\sqrt{2}} + 3 \cdot \frac{1}{\sqrt{2}} = \frac{2}{\sqrt{2}} + \frac{3}{\sqrt{2}} = \frac{5}{\sqrt{2}} \]

### Step 4: Find the component of \( \mathbf{A} \) along \( \hat{B} \)
The component of \( \mathbf{A} \) along \( \hat{B} \) is given by:
\[ \text{Component of } \mathbf{A} \text{ along } \hat{B} = \mathbf{A} \cdot \hat{B} \]
Thus, the component is:
\[ \text{Component of } \mathbf{A} \text{ along } \hat{B} = \frac{5}{\sqrt{2}} \]

### Final Answer
The component of vector \( \mathbf{A} \) along the vector \( \hat{B} \) is \( \frac{5}{\sqrt{2}} \).

---