Skip to main content
Pearson+ LogoPearson+ Logo
Start typing, then use the up and down arrows to select an option from the list.


Learn the toughest concepts covered in Physics with step-by-step video tutorials and practice problems by world-class tutors

3. Vectors

Adding Vectors by Components


Vector Addition By Components

Play a video:
Was this helpful?
Hey, guys. So previously we saw how to add vectors graphically, a situation like this where you have vectors A and B You want to calculate the resultant so you add them tip to tail and then you kind of a bunch of boxes. Well, sometimes, unfortunately, you're gonna have to calculate the resultant without being able to count squares because you're gonna have diagrams they don't have. They don't have grids. So in this video, we're going to see that all we just need our all of our vector math equations or triangle math equations, and we're gonna pull them all together to solve these kinds of problems. Now, these problems as we're gonna work one out together, are actually really, really repetitive. You end up doing the same things over and over and over again, so I'm gonna give you a list of steps that's gonna help you get the right answer every single time. Let's check it out. So this problem says that we're walking 5 m at 53 degrees, so that's a displacement, and then another displacement, another direction. We're gonna calculate the magnitude in the direction of the total displacement. Remember, total displacement could just be thought of as a resultant vectors. We're gonna be calculating a result in vector. All right, so the first step we're gonna do in these problems is we're gonna draw and then connect the vectors tip to tail. So when we're giving them graphically, we didn't have to draw them. But now that we don't have them, we're gonna have to draw them out because we don't have a grid. We're just kind of kind of sketch them out to the best of our ability. So we're gonna start the origin 5 m of 53. Look, something like this that looks about right. So there's five. We know this direction here, which is relative to the X axis, is 53. We've got eight at 30 so we connect them tip to tail. And this specter here, 30 degrees is a little bit shallower than 53 a little bit flatter. So it's gonna look something like this. So it's eight doesn't have to be perfect. And we know that this is 30 degrees above the X axis. Cool. So that's the first step. The second one is just we have to just draw out What? What the result in Vector is gonna look like So what? What is the resultant? Well, when we did this graphically, we just connect them to the tail. And the resultant was the shortest path from start to finish is basically is if we had actually just walked in this direction, the principle is the exact same. It's basically the shortest path and started the first one to the end of the last. And this is gonna be my results in vector. So this is gonna be the magnitude are and then it specified by an angle, which is they are. So we're trying to figure out the magnitude and the direction, which means that we're gonna need the components, remember how how do we solve for this? We take this resulting vector and we break it up into the triangles and we have to figure out what Rx and R. Y R. So we have to get these components over here when we do this Graphically, This was pretty straightforward. We have the magnitude. We break it up into its little a little triangles. We get Rx and R y. We could just count up the little boxes here. It was pretty straightforward. This was just four. And this is also just four. We don't have boxes in the situations. We can't calculate them, But just adding up all the boxes, there's nothing for us to count. We're gonna need a new method to figure out what these components are. So one way to think about how we got this Rx component is we can kind of break up the smaller vectors A and B into their small little triangles. So we can kind of think about this little triangle appear of this little X component. Basically, you just think about this a vector as breaking up into its triangles and we have this a X components, and then we also have BX component as well. And so we know this. A X component is three, which is kind of the boxes. You know, this be X component is just one. And so one way you can think about this is that this four is really just the addition of this three and this one put together. Now, this is gonna work the exact same way over here. So we have to do is we have to break up each of the vectors into the triangle. So this is my ex, and this is my A y. And then this is going to be my BX, and this is gonna be my B y. So now that leads us to the third step, which is we have to calculate what all of these components are. So that brings us to our equations. How do we take a vector, which we have the magnitude and the direction and calculate the components? We're just gonna use our vector decomposition equations. So that's what you are a cosine theta and are a sine theta equations. Now we have two vectors. This is a and this is gonna be be so the best way to keep track of all of the components is, uh I'm sorry I'm supposed to making that green. So the best way to keep track of all these components is by building a table. You're gonna have a lot of stuff floating around in your papers everywhere. It's gonna get organized. So I always highly encourage you guys build the table. So we want the A X components and the A Y components. So this we're just gonna use are a cosign data and a sign data equations. So my a vector, The magnitude is five. So this is gonna be five times the co sign of 53 degrees. And we're gonna get three. If you do the same thing for the Y direction, use sign. So five sign of 53 is for now. For the B vector, we do the same exact thing. Except the magnitude is eight. And the direction is 30 not 53. So we're gonna use eight times the cosine of 30 and you get 6.9 you eight times the sign of 30. You just get four. So now we know what all the components of, or all the tiny little legs of the triangles are. We know this is three. This is four. We know this is 6.9, and this is four. So, again, when we calculated when we figure out the resultant legs the legs of the Red Triangle here, we just added up the components of a X and B X. It's the same exact thing here. So this BX here we know is, you know, be X is 6.9. So if my if this leg here is three. And this leg here is 6.9. Then that means that my result in vector has to be the three and 6.9 added together. So on the table One way you can see this is that our result in vector is gonna be a plus b So we just add all the components straight down You're just gonna add everything vertically downwards three and 6.9 gives us 9.9 over here. So we know this is 9.9. And if you do the same thing for the Y Direction four and four is eight. So four in this direction and then four in this direction also make eight. It's obviously not to scale, so it doesn't look that way, but this is eight. So we're gonna just combine all of the X and Y components based on what this a Based on what? This our equation is, and in this example, it's a plus B. So that's really, really important. Sometimes it's not always gonna be a plus B, so you're just gonna always have to write out with our equation. Looks like Alright, guys. So now we have the legs of this result in Vector. So that leads us to the last step. Now we just have to figure out what the result with the magnitude and the direction are. And we have the components. So now that leads us to the other set of equations we're gonna use. We're trying to now go from the components and trying to figure the magnitude and direction. So now we're just gonna use our vector composition equations. Now, we're just gonna use our Pythagorean theorem and our tangent. Inverse, That's the last step. So our magnitude is gonna be the high pot news or the Pythagorean theorem of these components. So I've got 9.9 squared plus eight squared, and that just gives me 12.7. Uh, yeah, that's 12.7 now for the direction tha tha r remember. That's the angle relative to the X axis of our resulting vector. This is gonna be my tangent inverse of my white component, which is eight over my ex component, which is 9.9. And if you work this out, you're gonna get 38.9 degrees and these are your answers. So this is the magnitude and direction. That's how you do. Vector Edition. Follow the steps and you'll always get the right answer. That's it for this one. Let me know if you have any questions.

Adding Vectors

Play a video:
Was this helpful?
Hey, guys, let's check out this example together. So we've got these two vectors and A and B A has a magnitude of 10 and it's at degrees B has a magnitude three at 20 degrees, and we're gonna calculate the magnitude and direction off the resultant vector. But this result in Vector is not a plus B. It's a minus to be, so it's still gonna be vector addition. We just have to use a rules of subtraction and scaler multiples. So a minus to B is the same thing as saying a plus the negative of to be. So the first things first, we're just gonna draw and connect our vectors tip to tail. So we've got this vector A It's 10 at 40 degrees. So you start from the origin. So you've got this vector like this. This is gonna be a it's 10 and we know that this is a 40 degree angle. Now Victor B. Has a magnitude of three. The direction of 20. It's a little bit flatter. So we go from the tip to tail like this, and so this is gonna be RB vector and let's make a little bit shorter let's say that look like this. So we've got B is three. We know this angle here is 20 degrees s o. You know, hopefully you guys can see that. So now we have the vectors, we draw them tip to tail. But the problem is we have to remember that are resulting vector is not going to be from a to the end of B because we actually have to do a minus to be. So even though we've drawn out the vectors, they're actually still a little bit more than we need to do. So this result in vector first, we actually have to subtract to be here. So in order to subtract B, we're just gonna flip it. Basically, it's gonna go completely opposite direction with same length. So if be points in this direction, the negative B is gonna point exactly in the opposite direction. This is gonna be negative, B, and this is gonna be another negative be because we actually have to subtract to be so that means that my result in vector is again not from here to here. It actually you have to do a and you have to go minus to B and go in this direction. So my result in Vector is actually gonna be this vector over here. This is my our vector. So the second step here is we draw the resultant and its components. So that's the second step Rx and R. Why? So now we have to figure out the legs of this triangle over here. But, you know, again, So we have this vector here, we can calculate the components, but the resulting vector is actually if I go in this path over here, so let's go to the third step. So now, in order to calculate the legs of this, we have to break all the other triangles out into the components and then we have to calculate them. So this is my ex my A y. And then just for the sake of calculating the components of B X, we're gonna draw them out on this table over here or are in this diagram. So remember, we have to use a table in order to calculate all the X and Y components. So let's go ahead and do that. So I'm gonna get the table like this. This is my ex, and why components and then just for color coding, I'm just use This is my A and this is my B and then I'm gonna end up with a result in vector of R And then I have to write out the equation for our So let's go ahead and do that. So now if you want the X component of A we just have to stick to our vector decomposition equations. This is where we go to vectors from vectors, and then we want to decompose them into their components. So we just use our coastline data and a sign it data equations. So I'm gonna have for my ex component, I'm gonna use 10 and 40 degrees. And remember, as long as this is positive relative to the positive X direction and these equations gonna work So that means that my a co signed data here is gonna be 10 times the coastline of 40. When I end up with this 7.7, if you do the same thing for sign 10 times the sign of 40 degrees, what you get is 6.4. Now, if we do for B B cosigned data, that's this component over here So I know this is 7.7. This is 6.4. Might be cosine. Theta term is gonna be be times the cosine of 20 degrees. So this is gonna be three times the cosine of 20. And when I end up with his 2.8, so you know, this is 2.8 now, finally, three times the sign of 20 which is equal to one. So this is just one. All right, so these are all of our components. The table is really great at organizing all of this stuff. Now that links That brings us to the next step. If we want to calculate the result in vector, we're gonna have to combine all of these x and y components according to the our equation. I remember it's not a plus, B. You're not just gonna add all these components straight down, because remember, that is equal to a minus to be. What that means is that the X component of our is gonna be the X component of a minus two times the B X components. So if our is a minus to be, then Rx is a X minus two bx right just follows the same format. So my ex is gonna be 7.7 and then minus two times the X component, which is 2.8. If you work this out, your ex component is going to be 2.1. We do the exact same thing over here. This is gonna be a y minus to B y. So therefore, it's gonna be 6.1 minus two times one, and we get a white components of 4.4. So now we actually know what these are X and r Y components are. I know this is 2.1 and this is 4.4. And if you look at the diagram, it actually kind of makes sense. It's about 2.1 and 4.4. It's about twice as high. So this should make some sense to you. OK, so now let's last step. We just have to calculate our and tha r. And now we're going from the components of a vector. We have 2.1 and 4.4 the components, and now we actually want to construct the magnitude in the direction we use our Pythagorean theorem and our tangent inverse. So that means the magnitude of our it's just gonna be the Pythagorean theorem. We've got 2.1 squared, plus 4.4 squared. If you work this out, you're gonna get 4.9. So that is our results, since that's the magnitude of the results is now for the direction that's theta are This is gonna be the angle theta are over here. It's kind of messy, but I've got the tangent Inverse the arc tangent of my why component, which is my 4.4 over the X component, which is 2.1. So if you work this out, you're gonna get an angle of 64.5 degrees. So this over here, tha r is roughly 64.5 degrees. Alright, guys, that's it for this one. Let me know if you have any questions.