I need major help with this please!!!

On a safari, a team of naturalists sets out toward a research station located 6.9 km away in a direction 42° north of east. After traveling in a straight line for 2.1 km, they stop and discover that they have been traveling 22° north of east, because their guide misread his compass. What are the magnitude and direction (relative to due east) of the displacement vector now required to bring the team to the research station?

To find the magnitude and direction of the displacement vector required to bring the team to the research station, we can break it down into two components: the eastward component and the northward component.

Step 1: Eastward Component
The initial displacement in the eastward direction is given by:
eastward component = initial displacement * cos(angle)
eastward component = 2.1 km * cos(22°)

Step 2: Northward Component
The initial displacement in the northward direction is given by:
northward component = initial displacement * sin(angle)
northward component = 2.1 km * sin(22°)

These two components represent the displacement caused by the guide's error. Now, we need to find the displacement required to reach the research station.

Step 3: Final Displacement
To find the final displacement, we need to add the initial displacement to the displacement caused by the guide's error. This can be done using vector addition.

Let's use trigonometry to find the eastward and northward components of the final displacement.

Step 4: Eastward Component of Final Displacement
The eastward component of the final displacement is given by:
eastward component = eastward component (from initial displacement) + eastward component (from guide's error)
eastward component = 2.1 km * cos(42°) + 2.1 km * cos(22°)

Step 5: Northward Component of Final Displacement
The northward component of the final displacement is given by:
northward component = northward component (from initial displacement) + northward component (from guide's error)
northward component = 2.1 km * sin(42°) + 2.1 km * sin(22°)

Step 6: Magnitude of Final Displacement
The magnitude of the final displacement is given by the magnitude of the resultant vector, which can be found using the Pythagorean theorem:
magnitude = √((eastward component)^2 + (northward component)^2)

Step 7: Direction of Final Displacement
The direction of the final displacement can be found using trigonometry:
direction = atan(northward component / eastward component)

Now, let's plug in the values and calculate the final magnitude and direction:

eastward component = 2.1 km * cos(42°) + 2.1 km * cos(22°)
northward component = 2.1 km * sin(42°) + 2.1 km * sin(22°)
magnitude = √((eastward component)^2 + (northward component)^2)
direction = atan(northward component / eastward component)