A spelunker is surveying a cave. She follows a passage 180 m straight west, then 210 m in a direction 45° east of south, and then 280 m at 30° east of north. After a fourth displacement, she finds herself back where she started. Use a scale drawing to determine the magnitude and direction of the fourth displacement.

Video transcript

welcome back everybody. We have jane going on a scavenger hunt. So I am going to go ahead and draw our map and she's gonna go to a couple different places across the map. Now, just for convention sake, we're gonna have this be our X. Axis and this be our Y axis. So the first place that she travels to, she's going to head west and she is going to head west 150 m. I'm gonna call this displacement displacement a. From here. She is going to travel 240 m and this is at an angle That is 45° East from north. I'm gonna go ahead and call this displacement displacement. B. Makes another displacement. And she this time is going to walk 220 m. This is going to be 220 m. And she does this at an angle That is 60° east from south. And finally she is going to walk all the way back to the start and I'm gonna call this vector vector r. Sorry, I didn't label my previous vector, This could be vector C. And we are asked to find both the magnitude and direction of this final displacement. Well, since we're finding the magnitude and direction of the vector, we need some important formulas here. So we are going to have that the magnitude of a given vector will say vector a equal to the square root of its X. Component squared plus its Y component squared. We also have that. Its direction is given by the angle theta is the arc tangent of a Y over A X. But what are the X and Y components of our Well, she goes back to the origin right here. So we know that the final X is going to be zero and the final Y is going to be zero. Using this knowledge. We can actually add up all the X components and all the Y components to get that. So we're gonna have A X plus B X plus C X plus R. X equal to zero. And accordingly the same thing with B Y components where we have a Y plus B, Y plus C, Y plus R. Y is equal to zero. So let's go ahead and plug in what we know and find our our X and our Y. Alright, so X final x is equal to zero. This is going to be equal to well, A X. What is A X. Well, a only acts in the negative X direction. So this is just going to be negative plus the X component Of our b. So this is what we're looking for. BX. We know this angle right here so we can use that. So we're gonna have plus the magnitude of the vector to 40 times the sine of plus. We need our X component of C. So this is gonna be our C. X. And same thing here. We're going to have the magnitude of the vector, the sign of our angle Plus, our X is equal to zero. Simplifying this, we get that 510.2 plus our X is equal to zero, meaning that our X is equal to negative 10.2 m. So let's go ahead and do the same thing with Y now. So we can find our R Y, Y is equal to zero which is equal to the Y component of our A vector. Well A is only acting in the X direction, so it's not going to have a Y component plus the Y component of RB vector, which this is what we're looking for here. So same thing, we're gonna have the magnitude of the vector times this time the co sign of our angle plus R C Y. So we are going we are looking for this right here, RCY. This is going to be 220 times the co sign of 60. Now, this will be negative since we are headed in the downward directions plus our Y is equal to zero. Now, once again, let's simplify this and we are going to get 59. plus R Y is equal to zero, meaning that are Y is equal to negative 59.7 m. Now that we have our R X and r. Ry let's go ahead and use the above formulas to find our magnitude and direction. So the magnitude of our is going to be equal to the square root of its X. Component which is negative 5 10.2 squared was Y component to negative 59. squared. Plugging this into our calculator. We get 514 m now let's go ahead and find that direction. We have that our direction or angle Theta is equal to the arc tangent of negative or sorry the arc tangent of our Y. Component. 59.7 over our X component of 5. 10.2 negative 5 10.2. Now this is equal to 6.67°. But is this the angle that we're looking for? Let's take a closer look here. So the angle that we just found is actually this angle right here. But it is common practice to give an angle measure in terms of how far it is from the positive X axis. So we're actually gonna add 100 and 80 degrees to this to get our desired angle. So it's gonna be 6.67 plus 1 80 meaning that our data is equal to 186.7 degrees. So now we have found the magnitude of our final displacement and the direction of our final displacement which corresponds to answer voice. See Thank you guys so much for watching. Hope this video helped. We will see you all in the next one