A delivery truck travels 8 miles in the +x-direction, 5 miles in the +y-direction, and 4 miles again in the +x-direction. What is the magnitude (in miles) of its final displacement from the origin?

Hey, guys. Let's check out this practice problem just in case you need a little bit of help. We've got a delivery truck driver who's traveling in a bunch of different directions. So we've got 8 miles in the plus x, 5 miles in the plus y, and then 4 again in the plus x. We're gonna calculate the magnitude of the final displacement from the origin. Remember, that's just the resultant vector. So we're just gonna be adding all of these displacements, which we can think of as just, vectors. We're gonna connect them tip to tail, and then we're gonna figure out what the shortest path is. Let's get to it. So we've got this origin here. I'm gonna go 8 miles in the plus x direction. So I'm gonna go, each of these, you know, is worth 1 1 mile. So I'm gonna go 12345678. So I'm gonna end up over here. That's an arrow. So then for the next displacement, I just connect the arrow tip to tail. I wanna go 5 miles in the plus y direction. So you just go up from here. So I'm gonna go on the plus y. And then finally, the last displacement, again, connect it tip to tail, 4 miles again in the plus x direction. So that means I'm actually gonna go over here. 1, 2, 3, 4. So this is where I end up. Now you might think that the magnitude of the displacement is just 8, 5 and 4. You just add those up together. But remember, we're all just doing triangle math here and you can't connect these because these are perpendicular. The magnitude of the resultant or the final displacement is gonna be the shortest path from the start of the first to the end of the last one. So the my the my resultant vector over here, it's basically as if I I actually walked along the straight path from start to finish. I'm gonna call this resultant vector r and really what I'm trying to do is trying to figure out the magnitude of this vector here. So now that I've gotten this hypotenuse or I've gotten the shortest path, I can break it down into a triangle. That's what I always do with these vectors. So now I could just break it down into the legs which I can figure out. And so by just counting up all the little boxes, if each one of these squares is worth 1 mile, then I've got 5, 10 and then 12 over here. And then here, I've got 1, 2, 3, 4, 5. So now I've got the legs of the triangle. Now I just need to figure out the hypotenuse. That's just pythagorean theorem. So that means that the magnitude, which is given by these absolute value signs here, is just gonna be the square root of 12 squared plus 5 squared. And what you get is 13 miles. You don't get just the addition of 8, 5, and 4 because remember we're doing triangle math. So your resultant displacement or your final displacement from the origin is if you actually just walked 13 miles in this direction over here. Alright, guys. That's it for this one. Let me know if you have any questions.

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A delivery truck travels 8 miles in the +x-direction, 5 miles in the +y-direction, and 4 miles again in the +x-direction. What is the magnitude (in miles) of its final displacement from the origin?

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