Practice Finding Displacement Vectors

Instructor: Laura Foist

Laura has a Masters of Science in Food Science and Human Nutrition and has taught college Science.

Displacement is the distance from the starting point. In this lesson, we'll look at several examples of displacement problems and discuss how to solve them, particularly using the laws of sine and cosine.

Displacement vs. Distance

If you leave your home, walk all around town, stop at several stores, and then return home, your displacement is still 0 miles, even if you walked several miles that day! Displacement refers to how far away you currently are from your starting point (despite the path that you took).

Figuring out displacement is pretty straight forward when you are walking back and forth in straight lines, but it becomes increasingly complicated the more angles you add. The easiest way to think about displacement is, ''how far will I need to walk in order to walk directly back to my starting point?''

Displacement of a Line

Imagine you're sitting in the middle of an airplane, where you can only walk up or down the aisle. If you get up and walk 20 feet back to the bathroom, and then walk 30 feet forward past your seat, what is your current displacement from your seat?


Walking in an airplane


You walked 20 feet in the negative direction and then 30 feet in the positive direction. In order to determine displacement, add these two numbers together (including the positive and negative signs):

-20 + 30 = 10

You have been displaced 10 feet from your seat. Since the result is a positive number, you are 10 feet towards the front of the airplane.

Displacement in Blocks

Now imagine you're in a city organized into equal blocks with roads that extend north, east, south, and west. After starting out at home (yellow star), you walk:

  • 3 blocks east (red dot)
  • 2 blocks north
  • 1 block west (green dot)
  • 4 blocks south
  • 4 blocks west (blue dot)

What is your current displacement from home?


Blocks


The purple line indicates your current displacement. Notice that if we draw a line along the blocks we can form a right triangle with two blocks on each side:


showing triangle


Using the Pythagorean theorem, solve for x in order to find the displacement:


Pythagorean theorem


Substituting two blocks for a and two blocks for b, solve for x:


Solve


We can simplify this equation to solve for x:


Simplified equation


After simplifying the equation, we find that x = 2.8 blocks. Your displacement is 2.8 blocks from home.

Displacement and the Law of Cosine

Now let's say that you're out exploring with a compass. You first travel 10 miles 65 degrees east of north. From there you take a new bearing and travel 6 miles 60 degrees west of north. What is your displacement?


Displacement on a compass


You started out travelling 65 degrees east of north, which is the same as 25 degrees south of west at your new bearing: (90 - 75 = 15). Additionally, 60 degrees west of north is 30 degrees north of west (90 - 60 = 30). So the angle between the 10-mile line and the 6-mile line is 55 degrees (30 + 25 = 55):


Law of cos triangle


Solve for x using the law of cosines:


Law of cosine


In these examples, the lower case letters represent the sides, while the upper case letter represents the angle opposite that side.

Now that we know the length of sides a and b and the angle of C, we can find the length of side c. Let's plug in this information and solve for c:


Solve for length


We can simplify the equation to solve for c:


simplify c


After simplifying and solving, we find that the length of side c is 8.2 miles long.

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