an automobile travels 3 miles due north then 5 miles north of east. what is your resultant displacement?

North of east is not a direction

Do you mean Northeast? North by East? North North East ?

I will assume Northeast which is 45 degrees north of east

If so then:
North 3 + 5 sin 45 = 6.54
East= 0 + 5 cos 45 = 3.54

distance = sqrt (6.54^2 + 3.54^2)
= 7.43 miles
tan angle north of east = 6.54/3.54
angle north of east = 61.6 degrees
so bearing from origin clockwise from north = 90 - 61.6 = 28.4

Please correctly understanding.

To find the resultant displacement, we can use the Pythagorean theorem to calculate the magnitude of the displacement and trigonometry to find the angle.

Given:
Distance traveled due north = 3 miles
Distance traveled north of east = 5 miles

Step 1: Calculate the x and y components of the displacement:
The distance traveled due north is the y-component, and the distance traveled north of east is the x-component.

y-component (northward): 3 miles
x-component (eastward): 5 miles

Step 2: Calculate the magnitude of the displacement using the Pythagorean theorem:
Displacement^2 = (x-component)^2 + (y-component)^2

Displacement^2 = 5^2 + 3^2
Displacement^2 = 25 + 9
Displacement^2 = 34

Taking the square root of both sides:
Displacement ≈ √34

Step 3: Calculate the angle of the displacement:
Using trigonometry, we can find the angle using the following formula:
tan(θ) = (y-component)/(x-component)

Thus, tan(θ) = 3/5

Find θ:
θ ≈ arctan(3/5)

Using a calculator, θ ≈ 30.96 degrees

Therefore, the resultant displacement is approximately √34 miles in magnitude and it makes an angle of 30.96 degrees with the eastward direction.

To find the resultant displacement, we need to determine the magnitude (distance) and direction of the straight line from the starting point to the ending point.

First, let's visualize the two displacements:

1. The automobile travels 3 miles due north. This displacement is purely vertical. Draw a line pointing straight up (north) with a length of 3 miles.

2. The automobile then travels 5 miles north of east. This displacement is a combination of north and east directions. Draw a line 5 miles long at a 45 degree angle from the previous displacement. This line should be longer than the first line.

Now, let's find the resultant displacement:

1. Connect the starting point (end of the first displacement) with the ending point (end of the second displacement) using a straight line. This is the resultant displacement.

2. Measure the length of the straight line. This is the magnitude of the resultant displacement.

3. Measure the angle between the line and the north direction. This is the direction of the resultant displacement.

Using a ruler and protractor, you can determine that the length of the resultant displacement is approximately 7.81 miles, and the angle with respect to the north direction is approximately 26.6 degrees clockwise.

Therefore, the resultant displacement of the automobile is approximately 7.81 miles at an angle of 26.6 degrees clockwise from north.