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

Options

A. `(5)/(sqrt2)`
B. `10sqrt2`
C. `5sqrt2`
D. 5

Explanation

To find the component of vector \( \mathbf{A} = 2\hat{i} + 3\hat{j} \) along the vector \( \mathbf{B} = \hat{i} + \hat{j} \), we can 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 magnitude of vector \( \mathbf{B} \)
The magnitude of vector \( \mathbf{B} \) can be calculated using the formula:
\[ |\mathbf{B}| = \sqrt{(1)^2 + (1)^2} = \sqrt{1 + 1} = \sqrt{2} \]

### Step 3: Calculate the dot product \( \mathbf{A} \cdot \mathbf{B} \)
The dot product of vectors \( \mathbf{A} \) and \( \mathbf{B} \) is given by:
\[ \mathbf{A} \cdot \mathbf{B} = (2\hat{i} + 3\hat{j}) \cdot (\hat{i} + \hat{j}) = 2 \cdot 1 + 3 \cdot 1 = 2 + 3 = 5 \]

### Step 4: Calculate the component of \( \mathbf{A} \) along \( \mathbf{B} \)
The component of vector \( \mathbf{A} \) along vector \( \mathbf{B} \) can be calculated using the formula:
\[ \text{Component of } \mathbf{A} \text{ along } \mathbf{B} = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{B}|} \]
Substituting the values we found:
\[ \text{Component of } \mathbf{A} \text{ along } \mathbf{B} = \frac{5}{\sqrt{2}} \]

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

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