What is the component of `(3hati+4hatj)` along `(hati+hatj)` ?

Options

A. `(1)/(2)(hatj+hati)`
B. `(3)/(2) (hatj+hati)`
C. `(5)/(2)(hatj+hati)`
D. `(7)/(2)(hatj+hati)`

Explanation

To find the component of the vector \( \mathbf{A} = 3\hat{i} + 4\hat{j} \) along the vector \( \mathbf{B} = \hat{i} + \hat{j} \), we can use the formula for the component of one vector along another:

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

### Step 1: Calculate the dot product \( \mathbf{A} \cdot \mathbf{B} \)

Given:
- \( \mathbf{A} = 3\hat{i} + 4\hat{j} \)
- \( \mathbf{B} = \hat{i} + \hat{j} \)

The dot product \( \mathbf{A} \cdot \mathbf{B} \) is calculated as follows:

\[ \mathbf{A} \cdot \mathbf{B} = (3\hat{i} + 4\hat{j}) \cdot (\hat{i} + \hat{j}) = 3(\hat{i} \cdot \hat{i}) + 4(\hat{j} \cdot \hat{j}) + 3(\hat{i} \cdot \hat{j}) + 4(\hat{j} \cdot \hat{i}) \]

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

\[ \mathbf{A} \cdot \mathbf{B} = 3(1) + 4(1) = 3 + 4 = 7 \]

### Step 2: Calculate the magnitude squared of \( \mathbf{B} \)

The magnitude of \( \mathbf{B} \) is given by:

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

Thus, the magnitude squared is:

\[ |\mathbf{B}|^2 = (\sqrt{2})^2 = 2 \]

### Step 3: Substitute into the component formula

Now we can substitute the values into the component formula:

\[ \text{Component of } \mathbf{A} \text{ along } \mathbf{B} = \frac{7}{2} \mathbf{B} \]

Substituting \( \mathbf{B} = \hat{i} + \hat{j} \):

\[ \text{Component of } \mathbf{A} \text{ along } \mathbf{B} = \frac{7}{2} (\hat{i} + \hat{j}) = \frac{7}{2} \hat{i} + \frac{7}{2} \hat{j} \]

### Final Answer

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

\[ \frac{7}{2} \hat{i} + \frac{7}{2} \hat{j} \]