Question

In 2-d cartesian coordinates, a vector with magnitude 20 forms a 30 degree angle above the positive x-axis. for reference, the negative y-axis represents a 3/2π radian angle from the positive x-axis in this coordinate system. by what absolute value percent would the x & y components of this vector change if the vector's magnitude was doubled and the vector was rotated another 110 degrees counter-clockwise?

Answer 1

The Cartesian coordinates of the first vector are
20(cos(30°), sin(30°))
After the vector is multiplied by 2∠110°, its Cartesian coordinates are
40(cos(140°), sin(140°))

The x-component is larger than it was, but in the opposite direction. The percentage increase in the magnitude of the x-component is
x-change = (|40cos(140°)/(20cos(30°))| - 1)×100%
x-change ≈ 76.9%

The y-component is also larger than it was, but still in the +y direction. The percentage change in the magnitude of the y-component is
y-change = (|40sin(140°)/(20sin(30°))| -1)×100%
y-change ≈ 157.1%