## To determine the speed and direction of the airplane, we can break down the velocities into their horizontal and vertical components.

1. Draw a vector diagram: Start by drawing a diagram with a northward arrow representing the airplane's velocity of 100 mph. Then draw a vector representing the wind's velocity of 20 mph to the southeast (S45Â°E). Label each vector accordingly.

2. Calculate horizontal and vertical components: To find the components, use trigonometry. The angle between the direction of motion (north) and the wind (southeast) is 45 degrees.

- For the airplane's velocity, the horizontal component (north-south) is 100 mph, as it is heading due north. The vertical component (east-west) is 0 mph, as the airplane is not moving in an east-west direction.

- For the wind's velocity, the horizontal component is 20 mph*cos(45Â°) = 14.14 mph to the east (positive direction). The vertical component is 20 mph*sin(45Â°) = 14.14 mph to the south (negative direction).

3. Combine components to find the resultant velocity: Add the horizontal components and vertical components separately:

- Horizontal component: 100 mph + 14.14 mph = 114.14 mph to the north

- Vertical component: 0 mph - 14.14 mph = -14.14 mph to the south

4. Calculate the magnitude and direction of the resultant velocity: To find the overall speed, use the Pythagorean theorem:

- Magnitude: âˆš(114.14^2 + (-14.14)^2) = âˆš(13000 + 200) = âˆš13200 â‰ˆ 114.93 mph (rounded to two decimal places)

- Direction: To find the direction, use trigonometry. The angle with respect to the north direction (x-axis) can be determined as follows:

Î¸ = tan^(-1)((-14.14 mph) / (114.14 mph)) â‰ˆ -7.03 degrees (rounded to two decimal places)

Since the resultant velocity vector points in the northern quadrant, adding 180 degrees to the negative angle gives the direction:

Direction: -7.03Â° + 180Â° = 172.97Â° (rounded to two decimal places)

Therefore, the speed of the airplane is approximately 114.93 mph, and its direction is approximately 172.97 degrees relative to the north.