91Ó°ÊÓ

To solve problems involving motion, we often need to break a vector into its components. A vector has both magnitude (how long it is) and direction (where it points). In the case of the airplane, the wind acting as a vector has a speed (42 km/h) and direction (20° south of east). These vector components tell us exactly how much the vector affects movement in specific directions, like eastward or southward, using trigonometric functions.

We typically resolve the vector into two perpendicular components, one along the x-axis (eastward) and the other along the y-axis (northward), which involve:

• **Cosine** of the angle for the horizontal (eastward) component, since cosine relates an angle to the adjacent side of a right triangle.
• **Sine** of the angle for the vertical (southward or northward) component, since sine is linked to the opposite side of the angle.

By breaking vectors into components, we simplify complex movement into more manageable parts, allowing easier calculations.

Trigonometry is a branch of mathematics that studies the relationships between side lengths and angles of triangles, often employed to break vectors into components. In this exercise, we use trigonometric functions to resolve the wind's effect on the airplane. When we say the wind blows at 20° south of east, we can form a right triangle, where:

• The hypothenuse represents the total wind displacement.
• The eastward and southward movements are the triangle's sides.

We use:

• **Cosine function** to determine the eastward component: \( d_{east} = 10.5 \cdot \cos(20°) \).
• **Sine function** for the southward component: \( d_{south} = 10.5 \cdot \sin(20°) \).

This approach applies whenever an angle and hypotenuse are known, and we're identifying known sides, demonstrating the beauty of trigonometric relations.

The Pythagorean theorem is a fundamental principle in geometry, critical in calculating unknown sides or verifying distance in right triangles. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides.

Expressed mathematically, it reads as: \[ c^2 = a^2 + b^2 \]Where \( c \) is the hypotenuse, and \( a \) and \( b \) are the other two sides.
In our airplane problem, after calculating the northward and eastward velocity components (using 205.64 km/h and 9.86 km/h, respectively), we employ the Pythagorean theorem to find the resultant speed. This combination of speeds gives us the airplane's true velocity relative to the air:\[ v_{relative} = \sqrt{205.64^2 + 9.86^2} \approx 205.8 \text{ km/h} \]This method serves to accurately determine resultant velocities when vector components are perpendicular to one another.