Vector Addition Word Problems 2

Question

Ms. Sparks drives her car 35 miles at 15° south of west. There she drops off her math books and heads 65° south of east for 50 miles.

a) Make a head-to-tail drawing of the two vectors described.

b) Draw the resultant and find its magnitude and direction.

Answer 1

For a problem like this the drawing and the arithmetic are really the same thing, so let me turn each description into components first and the picture will follow.

For the first vector, 15° south of west means you start out pointing west and swing 15° toward south. West is the negative x direction and south is negative y, so

x₁ = −35 cos15° = −33.807
y₁ = −35 sin15° = −9.059

For the second, 65° south of east means start out pointing east and swing 65° toward south:

x₂ = 50 cos65° = 21.131
y₂ = −50 sin65° = −45.315

If you notice, the cosine always goes with the direction you started from and the sine goes with the direction you swung toward. That is the part students get backwards most often, so it is worth saying out loud each time instead of memorizing a formula.

For part a, the head to tail drawing is just those two arrows one after the other. Draw the first one from your starting point going down and to the left, only slightly below the west line since it is just 15°. Then start the second arrow at the tip of the first and send it down and to the right, steeply, 65° below the east line. The resultant is the single arrow drawn from the original starting point straight to the final tip.

For part b, add the components:

x = −33.807 + 21.131 = −12.676
y = −9.059 + (−45.315) = −54.374

Both came out negative, which means the resultant points into the southwest, and that should match what your drawing looks like. Always check that before going further, since a sign slip shows up here immediately.

The magnitude is just the Pythagorean theorem on those two legs:

R = √((−12.676)² + (−54.374)²) = √3117.23 = 55.83 miles

And the direction comes from the inverse tangent of the two legs:

tanθ = 54.374 / 12.676, so θ = 76.88°

We measured that angle from the west axis turning toward south, so our final answer is 55.83 miles at 76.88° south of west. The same arrow can also be described as 13.12° west of south, and many textbooks would write it as the bearing S 13.12° W. All three say exactly the same thing, so use whichever form your class uses.

As a check I found the magnitude a second way. The two directions differ by 100°, so in the head to tail triangle the angle between the 35 mile side and the 50 mile side is 80°, and the law of cosines gives √(35² + 50² − 2(35)(50)cos80°) = 55.83 miles as well. When two different methods land on the same number you can trust the components were set up right.