Video: Finding the Magnitude and Direction of a Displacement

A pedestrian walks 6.00 km east and then 13.0 km north. Find the pedestrian’s resultant displacement. At what angle north of east does the pedestrian walk.

05:12

Video Transcript

A pedestrian walks 6.00 kilometers east and then 13.0 kilometers north. Find the pedestrians resultant displacement. At angle north of east does the pedestrian walk.

In an exercise like this drawing a picture of what’s going on can be very helpful, so let’s draw a coordinate plane that shows the pedestrian’s motion. On this set of axes, we have north pointing up and east pointing right. Now let’s draw in the two legs of the pedestrian’s journey that they take.

If each tick mark on this plot represents three kilometers, then when our pedestrian moves 6.00 kilometers east, that will look like this arrow, this horizontal arrow. And the motion of 13.0 kilometers north will look like this. So in the end, our pedestrian is standing at this location and we want to find how far away that location is from our origin, and we also want to find what angle north of east does the pedestrian walk to get there.

So given the pedestrians final location, we want to discover what is the magnitude of the displacement; we’ll call it 𝑑. And we also want to discover what is the angle north of east that the pedestrian moves, and here you can see we’ve labeled that angle with a Greek letter called theta, so we want to solve for 𝑑 and 𝜃 in this example.

Now one great thing about this problem is that we’re working with a right triangle. That means the Pythagorean theorem applies to the situation. Recall what the Pythagorean theorem states. This theorem tells us that for a right triangle with sides 𝐴, 𝐵, and 𝐶, 𝐴 squared plus 𝐵 squared is equal to 𝐶 squared.

We can apply that to our particular problem by writing out 13.0 kilometers squared plus 6.00 kilometers squared equals 𝑑 squared, the displacement of our pedestrian. If we take the square root of both sides, then we see that 𝑑, the displacement of our pedestrian, equals the square root of 13.0 kilometers squared plus 6.00 kilometers squared.

When we type that into our calculator, we find that 𝑑 equals the square root of 206 kilometers squared, which altogether equals 14.3 kilometers. So that answer is part one; the displacement of a pedestrian is 14.3 kilometers. That’s the total distance the pedestrian traveled. Now we wanna answer part two, which is at what angle did the pedestrian walk.

In other words, we want to solve for 𝜃 in degrees. To do that, let’s take another look at this triangle the pedestrian created. Now first, we know that this is a right triangle, and we also know the lengths of all three sides of the triangle. So now let’s recall a relationship between those sides and the angles inside that triangle.

If we will get the ratio of the height of the triangle to the base length of the triangle, then we know there is an identity which says that the tangent of that angle we’ve called 𝜃 is equal to the opposite side of the triangle, in this case 13.0 kilometers, divided by the adjacent side, in this case 6.00 kilometers.

So let’s rewrite this identity in terms of our own numbers We can write that the tangent of our angle, 𝜃, is equal to 13.0 kilometers divided by six kilometers. And now if we take the arc tangent or the inverse tangent of both sides of our equation, we can see that on the left-hand side the arc tangent and the tangent function cancel one another and we’re left simply with 𝜃.

In other words, 𝜃 is equal to the inverse tangent of 13.0 kilometers, our vertical displacement, divided by 6.00 kilometers, our horizontal displacement. Typing this into our calculator, we find that in degrees, 𝜃 is equal to 65.2 degrees north of east. This is the angle at which the pedestrian will walk to end up at this final location.

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