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

Options

A. `(1)/(sqrt(2))`
B. `(3)/(sqrt(2))`
C. `(5)/(sqrt(2))`
D. `(7)/(sqrt(2))`

Answer

To find the component of vector \( \mathbf{A} = 2\hat{i} + 3\hat{j} \) along the vector \( \hat{i} + \hat{j} \), we will use the formula for the component of one vector along another vector. The formula is given by:

\[ \text{Component of } \mathbf{A} \text{ along } \mathbf{B} = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{B}|} \]

where \( \mathbf{A} \) is the vector we are analyzing, \( \mathbf{B} \) is the vector along which we want to find the component, and \( |\mathbf{B}| \) is the magnitude of vector \( \mathbf{B} \).

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

### Step 2: Calculate the dot product \( \mathbf{A} \cdot \mathbf{B} \)
The dot product of two vectors \( \mathbf{A} \) and \( \mathbf{B} \) is calculated as follows:

\[ \mathbf{A} \cdot \mathbf{B} = (2\hat{i} + 3\hat{j}) \cdot (\hat{i} + \hat{j}) \]

Using the properties of dot products:

\[ = 2(\hat{i} \cdot \hat{i}) + 2(\hat{i} \cdot \hat{j}) + 3(\hat{j} \cdot \hat{i}) + 3(\hat{j} \cdot \hat{j}) \]

Since \( \hat{i} \cdot \hat{i} = 1 \), \( \hat{j} \cdot \hat{j} = 1 \), and \( \hat{i} \cdot \hat{j} = 0 \):

\[ = 2(1) + 0 + 0 + 3(1) = 2 + 3 = 5 \]

### Step 3: Calculate the magnitude of vector \( \mathbf{B} \)
The magnitude of vector \( \mathbf{B} \) is calculated as:

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

### Step 4: Calculate the component of \( \mathbf{A} \) along \( \mathbf{B} \)
Now we can find the component of \( \mathbf{A} \) along \( \mathbf{B} \):

\[ \text{Component of } \mathbf{A} \text{ along } \mathbf{B} = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{B}|} = \frac{5}{\sqrt{2}} \]

### Final Answer
Thus, the component of vector \( \mathbf{A} \) along vector \( \mathbf{B} \) is:

\[ \frac{5}{\sqrt{2}} \]