Heidi T. answered • 09/09/19
MS in Mathematics, PhD in Physics, 7+ years teaching experience
To solve vector problems of this type, first define your coordinate system. How it is defined is up to you, but I will define it in the traditional manner with positive x to the east and positive y to the north. west is the negative of east and south is the negative of north. I always recommend drawing a picture. This forum doesn't allow me to show one, but try to draw it. Draw the vectors tail to tip in the order they are listed in the problem. The first is 2 units in the negative x direction. Then 8 in the negative y-direction, then up and positive in both x and y directions, the angle is 53 degrees with respect to the horizontal.
Next, need to break the problem down into components. You will consider components in the vertical and horizontal separately then combine them at the end. The first two legs of the journey are only on one direction, so are already in their components. The final leg is at an angle, so draw it as a right triangle, with a hypotenuse of 10 km and the 53 degree angle is between the horizontal leg and the hypotenuse.
Trig identities:
sin A = opp/hyp
cos A = adj/hyp
tan A = opp/adj
Using the trig identities, the components of the final leg are: horizontal = (10 km) cos (53); vertical = (10 km) sin (53)
Combining vectors:
Horizontal: ( -2 km) + (10 km) cos (53) = 4 km east (of the starting point)
vertical: (-8 km) + (10 km) sin (53) = 0 km north (of the starting point)
In this case, the final position is 4 km east of the starting point. We know it is east because the final answer is positive. If the vertical position had not been zero, we would have had to use the Pythagorean theorem to find the total distance from the origin and the inverse tangent equation to find the angle.
