IF `vec(A) = 5hat(i) + 12hat(j)` and `vec(B) = 3hat(i) + 4hat(j)`, find component of `vec(A)` in the direction of `vec(B)`.

Options

A. `(63)/(5)(5hat(i) + 12hat(j))`
B. `(32)/(25)(3hat(i) + 4hat(j))`
C. `(63)/(25)(3hat(i) + 4hat(j))`
D. `(63)/(25)(3hat(i) - 4hat(j))`

Answer

To find the component of vector \(\vec{A}\) in the direction of vector \(\vec{B}\), we can follow these steps:

### Step 1: Identify the vectors
Given:
\[ \vec{A} = 5\hat{i} + 12\hat{j} \]
\[ \vec{B} = 3\hat{i} + 4\hat{j} \]

### Step 2: Calculate the magnitude of vector \(\vec{B}\)
The magnitude of vector \(\vec{B}\) is calculated using the formula:
\[ |\vec{B}| = \sqrt{B_x^2 + B_y^2} \]
where \(B_x\) and \(B_y\) are the components of \(\vec{B}\).

Calculating:
\[ |\vec{B}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 3: Find the unit vector in the direction of \(\vec{B}\)
The unit vector \(\hat{b}\) in the direction of \(\vec{B}\) is given by:
\[ \hat{b} = \frac{\vec{B}}{|\vec{B}|} \]
Calculating:
\[ \hat{b} = \frac{3\hat{i} + 4\hat{j}}{5} = \frac{3}{5}\hat{i} + \frac{4}{5}\hat{j} \]

### Step 4: Calculate the dot product \(\vec{A} \cdot \hat{b}\)
The dot product is calculated as follows:
\[ \vec{A} \cdot \hat{b} = (5\hat{i} + 12\hat{j}) \cdot \left(\frac{3}{5}\hat{i} + \frac{4}{5}\hat{j}\right) \]
Calculating:
\[ \vec{A} \cdot \hat{b} = 5 \cdot \frac{3}{5} + 12 \cdot \frac{4}{5} = 3 + \frac{48}{5} = 3 + 9.6 = \frac{15}{5} + \frac{48}{5} = \frac{63}{5} \]

### Step 5: Find the component of \(\vec{A}\) in the direction of \(\vec{B}\)
The component of \(\vec{A}\) in the direction of \(\vec{B}\) is given by:
\[ \text{Component of } \vec{A} \text{ in the direction of } \vec{B} = (\vec{A} \cdot \hat{b}) \hat{b} \]
Calculating:
\[ \text{Component} = \frac{63}{5} \left(\frac{3}{5}\hat{i} + \frac{4}{5}\hat{j}\right) = \frac{63}{25} \cdot 3\hat{i} + \frac{63}{25} \cdot 4\hat{j} = \frac{189}{25}\hat{i} + \frac{252}{25}\hat{j} \]

### Final Result
Thus, the component of \(\vec{A}\) in the direction of \(\vec{B}\) is:
\[ \frac{189}{25}\hat{i} + \frac{252}{25}\hat{j} \]