If `vec(A) = 3hat(i) + 4hat(j)` and `vec(B) = 7hat(i) + 24hat(j)`, the vector having the same magnitude as `vec(B)` and parallel to `vec(A)` is :

Options

A. `5hat(i) + 20 hat(j)`
B. `15 hat(i) + 10 hat(j)`
C. `20 hat(i) + 15hat(j)`
D. `15hat(i) + 20 hat(j)`

Answer

To find the vector that has the same magnitude as \(\vec{B}\) and is parallel to \(\vec{A}\), we will follow these steps:

### Step 1: Find the unit vector of \(\vec{A}\)

The vector \(\vec{A}\) is given as:
\[ \vec{A} = 3\hat{i} + 4\hat{j} \]

To find the unit vector \(\hat{A}\), we first need to calculate the magnitude of \(\vec{A}\):
\[ |\vec{A}| = \sqrt{(3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

Now, the unit vector \(\hat{A}\) is given by:
\[ \hat{A} = \frac{\vec{A}}{|\vec{A}|} = \frac{3\hat{i} + 4\hat{j}}{5} = \frac{3}{5}\hat{i} + \frac{4}{5}\hat{j} \]

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

The vector \(\vec{B}\) is given as:
\[ \vec{B} = 7\hat{i} + 24\hat{j} \]

Now, we calculate the magnitude of \(\vec{B}\):
\[ |\vec{B}| = \sqrt{(7)^2 + (24)^2} = \sqrt{49 + 576} = \sqrt{625} = 25 \]

### Step 3: Construct the new vector \(\vec{C}\)

The new vector \(\vec{C}\) will have the same magnitude as \(\vec{B}\) and will be parallel to \(\vec{A}\). Therefore, we can express \(\vec{C}\) as:
\[ \vec{C} = |\vec{B}| \cdot \hat{A} \]

Substituting the values we found:
\[ \vec{C} = 25 \left( \frac{3}{5}\hat{i} + \frac{4}{5}\hat{j} \right) \]

Calculating this gives:
\[ \vec{C} = 25 \cdot \frac{3}{5}\hat{i} + 25 \cdot \frac{4}{5}\hat{j} = 15\hat{i} + 20\hat{j} \]

### Final Answer

Thus, the vector having the same magnitude as \(\vec{B}\) and parallel to \(\vec{A}\) is:
\[ \vec{C} = 15\hat{i} + 20\hat{j} \]

---