A light plane attains an air speed of \(480 \mathrm{~km} / \mathrm{h}\). The pilot sets out for a destination \(810 \mathrm{~km}\) to the north but discovers that the plane must be headed \(21^{\circ}\) east of north to fly there directly. The plane arrives in \(1.9 \mathrm{~h}\). What was the vector wind velocity?

Understanding the velocity vector components of an object, such as an airplane, is essential for analyzing its motion. Velocity, a vector quantity, means that it has both magnitude and direction. Breaking down the velocity into components allows for a detailed understanding of the movement in each independent direction, typically along the cardinal directions like east and west, or north and south.

For example, if a plane moves northeast, it has both northward and eastward components in its velocity. To find these components, trigonometry is used—specifically, the sine and cosine functions. The cosine of the angle gives the component in the direction of the original heading (in this case, northward), and the sine provides the component perpendicular to it (eastward). This concept was applied in Step 1 of our exercise, where the plane's velocity was broken into its northward and eastward components using a given angle.

Trigonometry plays a crucial role in physics, especially when dealing with vectors like velocity, force, or displacement. It helps in breaking down vectors into perpendicular components, which can then be independently analyzed and manipulated according to the laws of physics.

In the context of our exercise, trigonometric functions were used to determine the components of the plane's velocity. By multiplying the airspeed by \(\cos(21^\circ)\) and \(\sin(21^\circ)\), the northward and eastward components were calculated, respectively. The correct use of sine and cosine is vital for understanding vector quantities and provides a way to handle the directional nature of these physical quantities.

The resultant velocity is the combination of all velocity vectors acting on an object, providing a single vector that gives the overall motion. This is calculated by adding up all the individual velocity components corresponding to each direction. The principle of vector addition states that vectors must be added geometrically, taking into account both their magnitude and direction.

In our exercise, Step 2 required finding the effective velocity of the plane in a straight northward direction, ignoring any eastward or westward movement. This effective or 'resultant' velocity was determined simply by dividing the total northward distance by the time taken. Understanding resultant velocity is important in various contexts, such as navigating an airplane with wind influence or predicting the path of a projectile in physics.