A certain vector in the xy plane has an x-component of 12 m and a y-component of 8m. Its then rotated in the xy plane so that its x-component is halved. Then its new y-component is approximately :

Options

A. 14 m
B. 13.11 m
C. 10 m
D. `2.0` m

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Determine the Magnitude of the Original Vector
The original vector \( \vec{A} \) has components:
- \( A_x = 12 \, \text{m} \)
- \( A_y = 8 \, \text{m} \)

The magnitude of the vector \( |\vec{A}| \) can be calculated using the Pythagorean theorem:
\[ |\vec{A}| = \sqrt{A_x^2 + A_y^2} = \sqrt{12^2 + 8^2} \]
Calculating this:
\[ |\vec{A}| = \sqrt{144 + 64} = \sqrt{208} \]

### Step 2: Calculate the Magnitude
Now we simplify \( \sqrt{208} \):
\[ \sqrt{208} = \sqrt{16 \times 13} = 4\sqrt{13} \]

### Step 3: Determine the New X-component After Rotation
After rotation, the new x-component \( A'_x \) is given as half of the original x-component:
\[ A'_x = \frac{12}{2} = 6 \, \text{m} \]

### Step 4: Set Up the Equation for the New Magnitude
Since the magnitude of the vector remains the same before and after rotation, we can set up the equation:
\[ |\vec{A}| = |\vec{A}'| \]
This gives us:
\[ \sqrt{12^2 + 8^2} = \sqrt{(A'_x)^2 + (A'_y)^2} \]
Substituting the known values:
\[ \sqrt{208} = \sqrt{6^2 + A'_y^2} \]

### Step 5: Square Both Sides to Eliminate the Square Root
Squaring both sides:
\[ 208 = 6^2 + A'_y^2 \]
This simplifies to:
\[ 208 = 36 + A'_y^2 \]

### Step 6: Solve for the New Y-component
Rearranging the equation to solve for \( A'_y^2 \):
\[ A'_y^2 = 208 - 36 = 172 \]
Taking the square root to find \( A'_y \):
\[ A'_y = \sqrt{172} \]

### Step 7: Simplify the New Y-component
We can simplify \( \sqrt{172} \):
\[ \sqrt{172} = \sqrt{4 \times 43} = 2\sqrt{43} \]
Calculating \( \sqrt{43} \) gives approximately 6.56, thus:
\[ A'_y \approx 2 \times 6.56 \approx 13.12 \, \text{m} \]

### Conclusion
The new y-component after the vector is rotated is approximately:
\[ A'_y \approx 13.12 \, \text{m} \]