Nathaniel V. answered • 09/13/19
Aerospace (Aeronautical) Engineer Tutor
So you can solve this problem by setting up each individual contribution as a vector and then summing up the vectors to get the resultant. With the resultant vector, you can find the magnitude, which will be the distance from the starting location.
Givens:
1st Vector: 72 km in a direction 30° east of north
2nd Vector: 48 km due south
3rd Vector: 100 km 30° north of west
Assumption: We will assume that North is 90° to correspond to the Unit Circle or how a calculator defines the axis.
Equations:
Unit Vector: z = cos(theta) x + sin(theta) y
Solution:
You can start off by creating the vectors in terms of the unit vector by multiplying the magnitude to the unit vector
1st Vector: theta is 30 degrees closer to east from north therefore (90 - 30) deg
Z1 = 72km * [cos(90 - 30) x + sin(90 - 30) y] = 72km * [cos(60) x + sin(60) y] = 72km * [0.5 x + 0.866 y]
Z1 = 36km x + 62.352km y
2nd Vector: theta is due south therefore 270deg
Z2 = 48km * [cos(270) x + sin(270) y] = 72km * [0 x + -1 y]
Z2 = 0km x + -48km y
3rd Vector: theta is 30 degrees closer to north from west therefore (180 - 30) deg
Z2 = 100km * [cos(180-30) x + sin(180-30) y] = 100km*[cos(150) x + sin(150) y] = 100km*[-0.866 x + 0.5 y]
Z2 = -86.6km x + 50km y
Then you can sum up the vector x and vector y components:
Xtotal = 36 + 0 + -86.6 = -50.6 km
Ytotal = 62.352 + -48 + 50 = 64.352 km
Therefore the resultant vector is: Z = -50.6km x + 64.352km y
Then solving the square root of the sum of the squares, you can fins the magnitude:
Z = sqrt[(-50.6km)^2 + (64.352km)^2] = 81.86 km
Answer:
81.86 km from the starting point
