This problem involves determining the magnitude of the displacement vector for the aircraft traveling from point $A$ to point $C$, via $B$, using the given displacement triangle.

Solution:

1. Triangle Setup:

   • From the diagram:
     • $AB = 360 \, \text{km}$ (southward),
     • $BC = 100 \, \text{km}$ (eastward).

2. Displacement Vector $AC$: The total displacement from $A$ to $C$ forms the hypotenuse of a right triangle where:

   • The southward displacement ($AB$) is one leg of the triangle.
   • The eastward displacement ($BC$) is the other leg.

   Using the Pythagorean theorem: $AC = \sqrt{AB^2 + BC^2}$ Substitute the values: $AC = \sqrt{(360)^2 + (100)^2}$ $AC = \sqrt{129600 + 10000}$ $AC = \sqrt{139600}$ $AC = 374 \, \text{km} \, (\text{approx. to nearest whole number}).$

Final Answer:

The magnitude of the displacement from $A$ to $C$ is 374 km.

Would you like a deeper explanation or have other questions? 😊

Related Questions:

1. What is the direction (angle) of the displacement vector $AC$?
2. How can the vector $AC$ be represented in component form?
3. What would happen to the displacement if $BC$ were doubled?
4. How does the Pythagorean theorem work in this context?
5. What is the significance of displacement vs. total distance traveled?

Tip:

For problems involving displacement in two dimensions, always identify if the triangle is right-angled so you can apply the Pythagorean theorem.