What is the component of `3 hat(i) +4 hat(j)` along `hat(i)+hat(j)`:

Options

A. `(1)/(2)(hat(j) +hat(i))`
B. `(3)/(2)(hat(j) +hat(i))`
C. `(5)/(2)(hat(j) +hat(i))`
D. `(7)/(2)(hat(j) +hat(i))`

Answer

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 follow these steps:

### Step 1: Identify the vectors
Let:
- \( \mathbf{A} = 3\hat{i} + 4\hat{j} \)
- \( \mathbf{B} = \hat{i} + \hat{j} \)

### Step 2: Calculate the magnitude of vector \( \mathbf{B} \)
The magnitude of vector \( \mathbf{B} \) is given by:
\[ |\mathbf{B}| = \sqrt{(1)^2 + (1)^2} = \sqrt{2} \]

### Step 3: Find the unit vector in the direction of \( \mathbf{B} \)
The unit vector \( \hat{b} \) in the direction of \( \mathbf{B} \) is:
\[ \hat{b} = \frac{\mathbf{B}}{|\mathbf{B}|} = \frac{\hat{i} + \hat{j}}{\sqrt{2}} = \frac{1}{\sqrt{2}}\hat{i} + \frac{1}{\sqrt{2}}\hat{j} \]

### Step 4: Calculate the dot product of \( \mathbf{A} \) and \( \mathbf{B} \)
The dot product \( \mathbf{A} \cdot \mathbf{B} \) is:
\[ \mathbf{A} \cdot \mathbf{B} = (3\hat{i} + 4\hat{j}) \cdot (\hat{i} + \hat{j}) = 3 \cdot 1 + 4 \cdot 1 = 3 + 4 = 7 \]

### Step 5: Calculate the component of \( \mathbf{A} \) along \( \mathbf{B} \)
The component of \( \mathbf{A} \) along \( \mathbf{B} \) is given by:
\[ \text{Component of } \mathbf{A} \text{ along } \mathbf{B} = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{B}|} \hat{b} \]
Substituting the values we have:
\[ \text{Component of } \mathbf{A} \text{ along } \mathbf{B} = \frac{7}{\sqrt{2}} \left( \frac{\hat{i} + \hat{j}}{\sqrt{2}} \right) = \frac{7}{2} (\hat{i} + \hat{j}) \]

### Final Result
Thus, the component of \( \mathbf{A} \) along \( \mathbf{B} \) is:
\[ \frac{7}{2} (\hat{i} + \hat{j}) \]

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