91Ó°ÊÓ

A compass heading signifies the direction in which an aircraft is pointed, based on a compass. It is expressed in degrees, degrees relative to North. In the given problem, the airplane's compass heading is due west, equivalent to 270 degrees. Compass headings help pilots maintain their course by instantly showing directional changes.

Understanding compass headings involves being familiar with cardinal directions:

• North: 0 degrees
• East: 90 degrees
• South: 180 degrees
• West: 270 degrees

These directions form an important part of navigation and are vital for controlling aircraft paths. For the given scenario, having a wind from a specific direction affects the actual path, leading to a different resultant compass heading.

Vector addition is a process of combining two or more vectors to determine a resultant vector. Vectors have both magnitude and direction, which makes vector addition crucial in real-world applications like navigation.

In the original exercise, both the aircraft's velocity and the wind are vectors. The plane's velocity is pointed west, while the wind moves north. Thus, vector addition is employed to find the velocity of the airplane relative to the ground.

When performing vector addition:

• Place the tail of the second vector at the head of the first vector.
• The resultant vector is drawn from the tail of the first vector to the head of the last vector.
• This can also be done algebraically using components: add the respective i and j components.

For our problem, this results in a vector of \(-150i + 30j \mathrm{km/h}\)."

The Pythagorean Theorem is a fundamental principle in geometry, used here to find the magnitude of the resultant vector. Expressed as \(a^2 + b^2 = c^2\), where \(a\) and \(b\) are perpendicular components of a right triangle and \(c\) is the hypotenuse, or the vector's magnitude.

In this exercise, we treat the airplane's velocity and wind velocity as two sides of a right triangle:

• The westward airplane velocity represents one side: \(a = 150 \mathrm{km/h}.\)
• The northward wind velocity represents the other: \(b = 30 \mathrm{km/h}.\)

Using the theorem, the magnitude of the ground velocity \(c\) becomes \(|c| = \sqrt{150^2 + 30^2} = 152.5 \mathrm{km/h}.\)This gives a clear measure of the airplane's speed relative to the ground.

Calculating angles is vital in determining the direction of vectors. In our scenario, the angle of the resultant vector from the horizontal (westward) axis is calculated using the arctangent function.

The arctangent function relates the angle \(\theta\) to the ratio of the opposite side over the adjacent side of a right triangle. Here, we use the two vector components:

• The northward wind component: \(30\)
• The westward airplane velocity component: \(150\)

The formula used is: \(\theta = \tan^{-1}\left(\frac{30}{-150}\right).\)

This calculation results in an angle of -11.31 degrees relative to the negative x-axis (west direction).

When adjusting for standard compass notation, we subtract this from 180 degrees, giving us a heading of 168.69 degrees clockwise from east, a crucial element for navigation and ensuring accurate flight paths.