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

Options

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

Answer

To find the component of the vector \(3 \hat{i} + 4 \hat{j}\) along the vector \(\hat{i} + \hat{j}\), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Vectors**:
- Let \( \mathbf{V_1} = 3 \hat{i} + 4 \hat{j} \) (the vector we want to project).
- Let \( \mathbf{V_2} = \hat{i} + \hat{j} \) (the vector along which we want to project).

2. **Calculate the Magnitude of \( \mathbf{V_2} \)**:
- The magnitude of \( \mathbf{V_2} \) is given by:
\[ |\mathbf{V_2}| = \sqrt{(1^2 + 1^2)} = \sqrt{2} \]

3. **Calculate the Dot Product**:
- The dot product \( \mathbf{V_1} \cdot \mathbf{V_2} \) is calculated as follows:
\[ \mathbf{V_1} \cdot \mathbf{V_2} = (3 \hat{i} + 4 \hat{j}) \cdot (\hat{i} + \hat{j}) = 3 \cdot 1 + 4 \cdot 1 = 3 + 4 = 7 \]

4. **Calculate the Projection**:
- The projection \( P \) of \( \mathbf{V_1} \) along \( \mathbf{V_2} \) is given by:
\[ P = \frac{\mathbf{V_1} \cdot \mathbf{V_2}}{|\mathbf{V_2}|} = \frac{7}{\sqrt{2}} \]

5. **Express the Projection as a Vector**:
- To express the projection as a vector, we need the unit vector of \( \mathbf{V_2} \):
\[ \text{Unit vector along } \mathbf{V_2} = \frac{\mathbf{V_2}}{|\mathbf{V_2}|} = \frac{\hat{i} + \hat{j}}{\sqrt{2}} \]
- Therefore, the projection vector \( \mathbf{P} \) is:
\[ \mathbf{P} = P \cdot \text{Unit vector along } \mathbf{V_2} = \frac{7}{\sqrt{2}} \cdot \frac{\hat{i} + \hat{j}}{\sqrt{2}} = \frac{7}{2} \hat{i} + \frac{7}{2} \hat{j} \]

6. **Final Answer**:
- The component of \( 3 \hat{i} + 4 \hat{j} \) along \( \hat{i} + \hat{j} \) is:
\[ \frac{7}{2} \hat{i} + \frac{7}{2} \hat{j} \]