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 instructions:

### Step 1: Identify the initial components of the vector
The initial vector has an x-component of 12 m and a y-component of 8 m. We can represent this vector as:
\[ \vec{R_1} = 12 \hat{i} + 8 \hat{j} \]

### Step 2: Calculate the magnitude of the initial vector
The magnitude of the vector \(\vec{R_1}\) can be calculated using the Pythagorean theorem:
\[ |\vec{R_1}| = \sqrt{(12)^2 + (8)^2} = \sqrt{144 + 64} = \sqrt{208} \]

### Step 3: Determine the new x-component after rotation
After the vector is rotated, the x-component is halved. Therefore, the new x-component is:
\[ x' = \frac{12}{2} = 6 \text{ m} \]

### Step 4: Set up the new vector with the new x-component
The new vector after rotation can be represented as:
\[ \vec{R_2} = 6 \hat{i} + y \hat{j} \]
where \(y\) is the new y-component that we need to find.

### Step 5: Calculate the magnitude of the new vector
The magnitude of the new vector \(\vec{R_2}\) must be equal to the magnitude of the initial vector \(\vec{R_1}\) since the length of the vector remains unchanged during rotation. Thus:
\[ |\vec{R_2}| = \sqrt{(6)^2 + (y)^2} \]

### Step 6: Set the magnitudes equal to each other
Since the magnitudes are equal, we can write:
\[ \sqrt{(6)^2 + (y)^2} = \sqrt{208} \]

### Step 7: Square both sides to eliminate the square root
Squaring both sides gives:
\[ (6)^2 + (y)^2 = 208 \]
\[ 36 + y^2 = 208 \]

### Step 8: Solve for \(y^2\)
Rearranging the equation to solve for \(y^2\):
\[ y^2 = 208 - 36 = 172 \]

### Step 9: Calculate \(y\)
Taking the square root of both sides gives:
\[ y = \sqrt{172} \approx 13.11 \text{ m} \]

### Conclusion
The new y-component of the vector after rotation is approximately:
\[ y \approx 13.11 \text{ m} \]