How to Find a Displacement Vector

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  • 0:03 What Is a Displacement Vector?
  • 1:31 Steps for Displacement Vector
  • 4:08 The Alternate Method
  • 4:53 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

In this lesson, you'll learn how a displacement vector is totally different than the distance traveled. You'll learn how easy it is to find and add the displacement vectors of two different changes in an object's position.

What Is a Displacement Vector?

In this lesson, you'll learn how to add two displacement vectors. A displacement vector tells you how the position of an object has changed. This displacement vector includes not only how far you have traveled but also in which direction you have traveled. For example, say you start off at home and make your way to school. Your displacement vector starts at home and ends at your school. It's one straight line. It doesn't follow the path you took. On other hand, if you start at home and take a walk around the block with your dog, your end point is still at home, so therefore your displacement vector will be 0 since your beginning and end point position is the same. Displacement-wise, you didn't go anywhere.

This displacement vector can be drawn on the coordinate plane. The displacement vector then is given by the coordinate of the end point like this:

find displacement vector

So, when you went from home to school, your displacement vector as shown on the graph is (3, 4). Yes, you can write your displacement vector as a point on the coordinate plane as long as it begins at the origin. If these points are in miles, then this means that your school is located 3 miles to the east and 4 miles to the north. That is how much your position has changed.

Now, say you go from your school to the local ice creamery for a snack with your friends. You'll now have another displacement vector.

find displacement vector

This second displacement vector is (-1, 2). Remember, your displacement vectors always have a beginning at the origin.

To add these two displacement vectors, follow these steps:

Step 1: Find the coordinates of your two displacement vectors

If you are given a graph with no coordinates, then you'll first need to find the coordinate points of the ends of both displacement vectors when each displacement vector begins at the origin.

For your two displacements from home to school and from school to the ice creamery, you have already figured out the coordinate points of the displacement vectors. From home to school, it is (3, 4). From school to the ice creamery, it is (-1, 2).

Step 2: Move the second displacement vector so it starts where the first displacement vector ended

Next, you'll want to move your displacement vectors so they connect with each other. Where one ends, the other begins.

For your two displacement vectors (going from home to school and then from school to the ice creamery), you'll connect the displacement vector that begins at the school and ends at the ice creamery to the first displacement vector that begins at home and ends at the school.

find displacement vector

Step 3: Draw a new vector that is the addition of the two displacement vectors

This new vector will have the same beginning as your first displacement vector and end where your second displacement vector ends.

For your trip to school and then to the ice creamery, your new displacement vector will look like this:

find displacement vector

Your new displacement vector is the green vector. See how it begins where the first displacement vector begins, and it ends where the second displacement vector ends.

Step 4: Find the coordinates of the new displacement vector

You can find the coordinates by looking at your coordinate graph to see where the second displacement vector ends.

Looking at your graph, it looks like it ends at (2, 6).

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