A plane travels 1000 km at an angle of 40 degrees East to North, then changes direction and travels 500 km at an angle of 20 degrees North to East. What is the magnitude and direction of the planes total displacement?

I guess you mean north of east and east of north

east = 1000 cos 40 + 500 sin 20

north = 1000 sin 40 + 500 cos 20

east = 937
north = 1113

distance = sqrt(937^2+1113^2)

tan angle north of east = 1113/937

To find the magnitude and direction of the plane's total displacement, we can use vector addition.

Step 1: Break down the plane's displacement vectors into their components.

For the first leg of the journey (1000 km at an angle of 40 degrees East to North), we have:

Horizontal component: 1000 km * sin(40°) = 641.907 km (towards the east)
Vertical component: 1000 km * cos(40°) = 766.044 km (towards the north)

For the second leg of the journey (500 km at an angle of 20 degrees North to East), we have:

Horizontal component: 500 km * cos(20°) = 469.846 km (towards the east)
Vertical component: 500 km * sin(20°) = 171.01 km (towards the north)

Step 2: Add the horizontal and vertical components separately to find the total horizontal and vertical displacements.

Total horizontal displacement = 641.907 km + 469.846 km = 1111.753 km (eastward)
Total vertical displacement = 766.044 km + 171.01 km = 937.054 km (northward)

Step 3: Use the total horizontal and vertical displacements to find the magnitude and direction of the total displacement.

To find the magnitude of the displacement, we can use the Pythagorean theorem:

Magnitude of displacement = sqrt((total horizontal displacement)^2 + (total vertical displacement)^2)
= sqrt((1111.753 km)^2 + (937.054 km)^2)
≈ 1425.732 km

To find the direction of the displacement, we can use trigonometry:

Direction = arctan(total vertical displacement / total horizontal displacement)
= arctan(937.054 km / 1111.753 km)
≈ 40.37°

Therefore, the magnitude of the plane's total displacement is approximately 1425.732 km, and the direction is approximately 40.37° (measured from east, counterclockwise).