# Help me find the magnitute & direction of her displacement: A very strong girl runs 200 miles east for several hours. After a brief rest she. . .travels 100 miles north. She stops for the night...

Help me find the magnitute & direction of her displacement: A very strong girl runs 200 miles east for several hours. After a brief rest she. . .

travels 100 miles north. She stops for the night at a local bed & breakfast & then she travels 75 miles directly south-east. Find the magnitute and direction of her displacement. I have no clue how to solve this problem & any help would be greatly appreciated. God bless you.

ndnordic | High School Teacher | (Level 2) Associate Educator

Posted on

The easiest way to solve vector problems of this type is to break each vector into its x- and y- components and then add the individual x and y values to find the final displacement. Once you have the final point you can use trig functions to find the angle of the displacement.

You also need to establish what values are positive and what values are negative. Generally, to the east and north are considered positive and to the west and south are considered negative.

So:

vector one has a y value of 0 and an x value of +200 miles

vector two has a y value of + 100 miles and an x value of 0

for vector three  you know the angle of travel is southeast  so that corresponds to an angle of 45 degrees south of east. So construct a 45,45,90 right triangle where the 75 miles is the hypotenuse. You are traveling both south (negative direction) and east (positve direction) so keep this in mind when you solve for the individual vectors.

The sin 45 = x/75, so x = 75 * sin 45 = +53.03 miles

The cos45 = y/75, so y = 75 * cos 45 = -53.03 miles

Now add all the vector components:

In the x-direction you have +200, 0, and +53.03 miles = 253.03 miles east.

In the y-direction you have 0, +1000, - 53.03 = + 46.97 miles north.

So your final position is 253.03 miles east and 46.97 miles north. So to see this, draw a right triangle starting at the origin, going 253.03 miles east, the north 46.97 miles.

Use Pythagorean Theorem to find the straight line distance:

253.03^2 + 46.972= z^2

z= 257.35 miles.

To find the angle, use the inverse tangent function.

tan^-1 (46.97/253.03) = 10.52 degrees north of east.

So your displacement is 257.35 miles at an angle of 10.52 degrees north of east.