B

Step 1: Identify the vector components. The vector shown can be represented in component form as \( extbf{A} = (A_x, A_y) \) where \( A_x = 3 \) and \( A_y = 4 \).
Step 2: Calculate the magnitude of the vector: \( | extbf{A}| = \sqrt{A_x^2 + A_y^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \).
Step 3: Rotate the vector by 120º counterclockwise. Using the rotation matrix, the new components \( (A'_x, A'_y) \) become:
\( A'_x = A_x \cdot \cos(120º) - A_y \cdot \sin(120º) \)
\( A'_y = A_x \cdot \sin(120º) + A_y \cdot \cos(120º) \).
Substituting values, \( \cos(120º) = -0.5 \) and \( \sin(120º) = \frac{\sqrt{3}}{2} \):
\( A'_x = 3 \cdot (-0.5) - 4 \cdot \left(\frac{\sqrt{3}}{2}\right) = -1.5 - 2\sqrt{3} \)
\( A'_y = 3 \cdot \left(\frac{\sqrt{3}}{2}\right) + 4 \cdot (-0.5) = \frac{3\sqrt{3}}{2} - 2 \).
Step 4: Halve the magnitude of the new vector:
The new vector now becomes \( \textbf{B} = (\frac{1}{2} A'_x, \frac{1}{2} A'_y) \).
Therefore, combining steps gives the resulting vector as chosen option B.