If `vecA=3hati+4hatj` and `vecB=7hati+24hatj`, find a vector having the same magnitude as `vecB` and parallel and same direction as `vecA`.

Options

A. `5hati+20hatj`
B. `15hati+10hatj`
C. `20hati+15hatj`
D. `15hati+20hatj`

Explanation

To solve the problem, we need to find a vector that has the same magnitude as vector B and is parallel to vector A. Let's go through the steps systematically.

### Step 1: Identify the vectors
We have:
- \( \vec{A} = 3\hat{i} + 4\hat{j} \)
- \( \vec{B} = 7\hat{i} + 24\hat{j} \)

### Step 2: Calculate the magnitude of vector B
The magnitude of a vector \( \vec{B} = a\hat{i} + b\hat{j} \) is given by the formula:
\[ |\vec{B}| = \sqrt{a^2 + b^2} \]
For vector B:
\[ |\vec{B}| = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25 \]

### Step 3: Calculate the magnitude of vector A
Using the same formula for vector A:
\[ |\vec{A}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 4: Determine the scaling factor
We need a vector that has the same magnitude as vector B (which is 25) and is in the same direction as vector A. Since the magnitude of A is 5, we need to scale vector A by a factor of:
\[ \text{Scaling factor} = \frac{|\vec{B}|}{|\vec{A}|} = \frac{25}{5} = 5 \]

### Step 5: Scale vector A
To find the new vector \( \vec{R} \), we multiply vector A by the scaling factor:
\[ \vec{R} = 5 \cdot \vec{A} = 5 \cdot (3\hat{i} + 4\hat{j}) = 15\hat{i} + 20\hat{j} \]

### Step 6: Final Result
The vector that has the same magnitude as \( \vec{B} \) and is parallel to \( \vec{A} \) is:
\[ \vec{R} = 15\hat{i} + 20\hat{j} \]

### Summary
Thus, the final answer is \( \vec{R} = 15\hat{i} + 20\hat{j} \).

---