A and B are two vector given by `A= 2hati +3hatjand B=2hati+4hatj` The magnitude to the component of A along B is

Options

A. `(5)/(sqrt(2))`
B. `(3)/(sqrt(2))`
C. `(8)/(sqrt(5))`
D. `(5)/(sqrt(13))`

Explanation

To find the magnitude of the component of vector A along vector B, we will follow these steps:

### Step 1: Define the Vectors
Given:
- Vector A = \( 2 \hat{i} + 3 \hat{j} \)
- Vector B = \( 2 \hat{i} + 4 \hat{j} \)

### Step 2: Calculate the Dot Product of A and B
The dot product \( A \cdot B \) is calculated as follows:
\[ A \cdot B = (2 \hat{i} + 3 \hat{j}) \cdot (2 \hat{i} + 4 \hat{j}) \]
Using the properties of dot products:
\[ A \cdot B = (2)(2) + (3)(4) = 4 + 12 = 16 \]

### Step 3: Calculate the Magnitude of Vector B
The magnitude of vector B is given by:
\[ |B| = \sqrt{(2)^2 + (4)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5} \]

### Step 4: Calculate the Component of A Along B
The component of vector A along vector B is given by the formula:
\[ \text{Component of A along B} = \frac{A \cdot B}{|B|} \]
Substituting the values we found:
\[ \text{Component of A along B} = \frac{16}{2\sqrt{5}} = \frac{16}{2} \cdot \frac{1}{\sqrt{5}} = \frac{8}{\sqrt{5}} \]

### Step 5: Final Result
Thus, the magnitude of the component of vector A along vector B is:
\[ \frac{8}{\sqrt{5}} \]

### Conclusion
The correct answer is \( \frac{8}{\sqrt{5}} \).

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