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Understanding vector addition is crucial when solving problems involving relative velocity. Vectors help us represent quantities that have both magnitude and direction, such as velocity, which cannot be adequately described by just a single number.

In physics, vector addition is used to combine multiple vectors into a single vector. For example, the airplane's actual motion is a result of adding its intended velocity vector (the direction the pilot sets the course) to the wind velocity vector. When an airplane aims to travel west but ends up slightly south, the pilot experiences how these velocities interact.

To add vectors, you must align them based on their direction and ensure their tails start from the same point. If you then draw a new vector from the start point to the end point of the final vector, you'd have a clear representation of the plane's actual path. This technique helps visualize how different forces (or in this case, velocities) interact in determining resultant directions.

The result is fundamental in problems like this: calculating the wind's velocity or adjusting the plane's course to compensate for wind. By breaking down movements into components using vector addition, students can understand and solve such complex scenarios more easily.

The Pythagorean Theorem is a helpful tool in calculating resultant distances when dealing with vectors that form right-angled triangles. Often in physics, two different directional vectors combine to form a new one at an angle, and deciphering this resultant vector is where the Pythagorean Theorem shines.

In our airplane problem, the movements west (120 km) and south (20 km) form the legs of a right triangle. The theorem states that for a right triangle with legs of length 'a' and 'b', and hypotenuse 'c', the relation is: \( c^2 = a^2 + b^2 \).

Applying the Pythagorean Theorem allows us to find the exact path's length, in this case, the actual distance the plane traveled, calculated as: \( V_a = \sqrt{{(120 km)^2 + (20 km)^2}} \). This results in the hypotenuse of the triangle, and hence the magnitude of the total journey described by the path.

By understanding the theorem, students can break down complex travel problems into more manageable components, making it easier to solve part by part.

Trigonometry helps us determine angles and side lengths in scenarios involving right triangles, crucial in solving real-world physics problems. When you have a situation like a plane's course being altered by wind, trigonometry comes in handy to find out exactly how much the direction will change.

In the airplane problem, after forming a right triangle with the actual path west and south, we apply tangent trigonometry to determine the angle: \( \tan \theta = \frac{\text{southward distance}}{\text{westward distance}} = \frac{20 km}{120 km} \).

Solving this gives us \(\theta\), the angle by which the plane deviates from its intended westward direction. Trigonometric identities like sine, cosine, and tangent allow us to connect angle measurements with side lengths in right triangles, translating into understanding real-world changes in direction.

For adjusting the course when the wind is known, the inverse tangent (\(\tan^{-1}\)) is applied when rediscovering or correcting the course needed to maintain a due west heading despite a southward wind blow. This demonstrates trigonometry’s elegant application beyond pure mathematics, providing clarity in path predictions and course adjustments.