The components of the vectors are:
- \(\vec{A} = (20 \cos(40^\circ), 20 \sin(40^\circ))\)
- \(\vec{B} = (-15 \cos(30^\circ), 15 \sin(30^\circ))\)
- \(\vec{C} = (-3, 4)\)
To find the angle between \(\vec{A}\) and \(\vec{C}\), use the dot product formula:
\[
\vec{A} \cdot \vec{C} = A_x C_x + A_y C_y
\]
\[
|\vec{A}| |\vec{C}| \cos(\theta) = \vec{A} \cdot \vec{C}
\]
\[
\cos(\theta) = \frac{\vec{A} \cdot \vec{C}}{|\vec{A}| |\vec{C}|}
\]
\[
\theta = \cos^{-1}\left(\frac{\vec{A} \cdot \vec{C}}{|\vec{A}| |\vec{C}|}\right)
\]
Substitute the values to find the angle \(\theta\).