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To solve this problem, we need to use vector addition to account for both the plane's airspeed and the wind's effect on the direction.

Here's the breakdown:

1. **Define the vectors**:
   - The plane's airspeed vector (\(V_p\)) is directed at an unknown angle \( \theta \) east of north, with a magnitude of 700 km/h.
   - The wind vector (\(V_w\)) is coming from the northeast at an angle of 45° from the north (or east), with a magnitude of 40 km/h.

2. **Set up vector components**:
   - The goal is for the resultant vector (combination of the plane's velocity and wind's effect) to point directly north.
   - This means that the east-west component of the resultant vector should be zero.

3. **Decompose the wind vector**:
   - Since the wind is from the northeast, it has both northward and westward components:
     \[
     V_{w, x} = 40 \cos(45^\circ) = 40 \times \frac{\sqrt{2}}{2} = 20 \, \text{km/h (eastward)}
     \]
     \[
     V_{w, y} = 40 \sin(45^\circ) = 40 \times \frac{\sqrt{2}}{2} = 20 \, \text{km/h (northward)}
     \]

4. **Set up the plane's velocity components**:
   - The plane's velocity vector (\(V_p\)) should counter the eastward component of the wind to ensure it travels directly north.
   - Let \(\theta\) be the angle east of north that the plane needs to fly. Then:
     \[
     V_{p, x} = 700 \sin(\theta)
     \]
     \[
     V_{p, y} = 700 \cos(\theta)
     \]

5. **Set up the equation for no eastward component**:
   - To cancel out the wind’s eastward component, we set:
     \[
     700 \sin(\theta) = 20
     \]
   - Solving for \(\theta\):
     \[
     \sin(\theta) = \frac{20}{700} = 0.02857
     \]
     \[
     \theta = \arcsin(0.02857) \approx 1.64^\circ
     \]

So, the pilot should fly approximately **1.64 degrees east of north** to maintain a due north course.

Would you like more details on any step, or have additional questions?

Here are some related questions to explore further:

1. What would the required heading angle be if the wind speed were doubled?
2. How does the angle change if the plane's airspeed is increased?
3. What if the wind direction changes to due east? How would that affect the angle?
4. How can this problem be solved using vector diagrams?
5. What role does trigonometry play in navigation for pilots?

**Tip**: When dealing with navigation problems, setting up a clear diagram with vectors can help visualize the solution.

An airplane needs to head due north, but there is a wind blowing from the northeast at 40 km/hr. The plane flies at an airspeed of 700 km/hr. To end up due north, the pilot will need to fly the plane ___ degrees east of north.

Learn how to calculate the heading angle for a plane to counteract wind blowing from the northeast. With an airspeed of 700 km/h and a wind speed of 40 km/h, find the angle east of north to maintain a due north course. This solution involves vector addition and trigonometry, ideal for Grades 11-12.

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