A plane moves due east while the pilot points the plane somewhat south of east, toward a steady wind that blows to the northeast. The plane has velocity `vecv_(PW)` relative to the wind, with an airspeed (speed relative to the wind) of 215 km/h, directed at angle `theta ` south of east. The wind has velocity ` vecv_(WG)` is relative to the ground with speed 65.0 km/h, directed `20.0^@` cast of north. What is the magnitude of the velocity `vec_(PG)` of the plane relative to the ground, and what is `theta ` ?

Explanation

To solve the problem, we need to analyze the motion of the plane and the wind in two dimensions. We will break down the velocities into their components and use trigonometric relationships to find the angle \( \theta \) and the magnitude of the velocity of the plane relative to the ground.

### Step 1: Define the velocities
1. The velocity of the plane relative to the wind is given as:
\[ \vec{v}_{PW} = 215 \text{ km/h} \text{ at an angle } \theta \text{ south of east} \]
This can be broken down into components:
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