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 use the formula for the projection of one vector onto another. The formula is given by:

\[ \text{Component of } \vec{A} \text{ in the direction of } \vec{B} = \frac{\vec{A} \cdot \vec{B}}{|\vec{B}|^2} \vec{B} \]

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

Given:
\[ \vec{A} = 5\hat{i} + 12\hat{j} \]
\[ \vec{B} = 3\hat{i} + 4\hat{j} \]

The dot product \(\vec{A} \cdot \vec{B}\) is calculated as follows:
\[ \vec{A} \cdot \vec{B} = (5\hat{i} + 12\hat{j}) \cdot (3\hat{i} + 4\hat{j}) = 5 \cdot 3 + 12 \cdot 4 = 15 + 48 = 63 \]

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

The magnitude of \(\vec{B}\) is given by:
\[ |\vec{B}| = \sqrt{(3^2 + 4^2)} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 3: Calculate \(|\vec{B}|^2\)

Now, we calculate \(|\vec{B}|^2\):
\[ |\vec{B}|^2 = 5^2 = 25 \]

### Step 4: Substitute values into the projection formula

Now we substitute the values into the projection formula:
\[ \text{Component of } \vec{A} \text{ in the direction of } \vec{B} = \frac{63}{25} \vec{B} \]

Substituting \(\vec{B}\):
\[ \text{Component of } \vec{A} \text{ in the direction of } \vec{B} = \frac{63}{25} (3\hat{i} + 4\hat{j}) \]

### Step 5: Distributing the scalar

Distributing \(\frac{63}{25}\):
\[ \text{Component of } \vec{A} \text{ in the direction of } \vec{B} = \left(\frac{63 \cdot 3}{25}\right) \hat{i} + \left(\frac{63 \cdot 4}{25}\right) \hat{j} \]
\[ = \frac{189}{25} \hat{i} + \frac{252}{25} \hat{j} \]

### Final Answer

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