 Let's solve a question on adding vectors using the component method. We can see that this one is a v vector, this one is w. The magnitude of vector v is 6, magnitude of vector w is 4, and vector v is making an angle of 100 degrees from the positive x-axis, w is making an angle of 40 degrees from the positive x-axis. First part is to figure out v plus w, and we have to show that with their i and j cap components. And the second part is to figure out the magnitude of the resultant and the angle with the positive x-axis. So as always, why don't you pause this video first, and try to attempt both the parts on your own first. Alright, hopefully you have given this a shot. Now let me hide the second part first, we will come to the magnitude of the resultant and the angle, but first let's see what the resultant actually looks like. So let me hide this one for now. Let's just focus on part one. Here we need to add vector v and vector w, and we will use the component method for that. So one thing that we already know is that, let's say if we have this vector, a vector component describes the effect of this vector in that direction. So let's say, so if we have a vector oriented like this, it can have a horizontal component which looks like this. So this blue horizontal component is showing the effect of the red vector in the horizontal direction, and this green vertical component, this one is showing the effect of this red vector in the vertical direction. So the vector components, they really show the effect of the vector in that particular direction. And for any angle vector, vector at any angle, it will have a horizontal component and a vertical component. So let's say if you have some vector that looks, let's say it looks like this. And this one will have a horizontal component that could look somewhat like this, and a vertical component, vertical component like this, so it will be slightly longer, alright. So any angle vector will have two components, and we can figure out the magnitude of these components. We know how to do that, right? We need to know two things. If we know the magnitude, if we know the magnitude and the direction of the vector, the direction of, in this case, the red vector. We can work out the magnitude of the horizontal and the vertical components. For this case, if let's say this angle is theta, then the horizontal component, let's say the magnitude of this vector is A, then the horizontal component that would be A cos theta. The component that is adjacent to theta is the cos component. The component that is opposite to theta, this one, this one would be A sin theta. And we can even add two angle vectors just by looking at their component. So let's say if we have one more vector angle to this red vector, and we want to add it, and let's say that blue vector looks somewhat like this. So all we need to do, all we need to do is even resolve this smaller lighter blue vector into its vertical and horizontal component. This is how the vertical component would look like, and this is how the horizontal component would look like. And now we can just look at the, we can just subtract the blue horizontal components, and the vertical components are in the same direction, so they can be added. So we can find the components of the resultant vector. And then using Pythagoras, we can even find its magnitude. So the great thing about this method is that we don't really need to know, we don't really need to remember any formula or do any graphical version of adding vectors of anything of that sort. We just need to know how to resolve vectors into their components. That's all. Using just that, we can figure out the resultant of any two vectors. So we will use this strategy. We will use the strategy of resolving vector into their components, adding their corresponding i-caps and j-caps, then figuring out the final magnitude and its angle with the positive x-axis. Okay, now let's come back to this one. Let's look at v and w independently. Let's try and resolve them into their components and see how they look like. Let me make this disappear. So this blue one, this blue one will have one horizontal component and that will look, that will look, let's say like this. And it will have a vertical component. I'm showing all the horizontal with this color and all the vertical components with a green color. So this is its vertical component. So firstly, we are writing vector v into component form. So for that, we would need, this would be ax i-cap plus ay j-cap and ax is the horizontal component. So we know that the, we know that vector v makes an angle of 100 from the positive x-axis. That means that this angle right here, this angle, that would be 80 degrees, 180, 180, minus 100. And the component adjacent to 80, that would be 6 cos 80 because it's adjacent to theta, that would be cos component. So ax, ax here is, this right here is 6 into cos 80 degrees. And the vertical component, that would be 6 sin 80 degrees. So this is 6 sin 80 degrees. The component opposite to the angle, that would be sin component. So it's 6 sin 80. But there's one more thing. We see that the horizontal component, it's pointing in the negative x direction. So there will be a negative sign over here as well, because it's pointing in the negative x direction. Now, if you work this out, minus 6 cos 80, this would be minus 1.04 i-cap plus 5.90 j-cap. So this is your vector v. Now let's work out vector w, vector w, again vector w, we are resolving it into components and we want to write it in the same manner as we wrote vector v, so in its component format. So this is axi plus ayj. So for vector w, horizontal component can look somewhat like this, slightly more. And the vertical component, that would be like this. This is your vertical component. So horizontal component, that is 4, the magnitude of vector w, 4 into cos 40 degrees. So this vector is already making an angle of 40 with the positive x axis and the component adjacent to 40, adjacent to theta, is the cos component. So ax, this is 4 into cos 40 degrees i-cap plus ay, the vertical component that is 4 into sin 40 degrees j-cap. And when we work this out, this is 3.06 i-cap plus 2.57 j-cap. This is vector w. And now we can add the x or the i-components of both to give us a total x-component. So we can add, we can add, we can add this x-component with this x-component. And we can add this j-component with this j-component. So when we do that, vector v plus vector w, this is equal to minus 1.04 plus 3.06, that is just 2 i-cap plus 5.90 plus 2.57. That is 8.47 j-cap. So v plus w, this is 2 i-cap plus 8.47 j-cap. So this was the first part. And in the second part, we are supposed to figure out the magnitude of the resultant vector and the angle that it makes with the positive x-axis. So let's do that. We can use the Pythagoras theorem to figure out the magnitude of the resultant vector because we already know it's x and y-component. So this vector really, this vector really kind of looks like this. It has a horizontal x-component of magnitude 2 and it has a vertical y-component. And the magnitude of this is 8.47. So by using Pythagoras theorem, 2 square plus 8.47 square root of the sum, that will give us a magnitude. So when we do that, when we do that, if we write the magnitude, let's write that over here. The magnitude, that would be under root of 2 square plus 8.47 whole square. And when you work this out, this comes out to be equal to 8.47. The angle with the positive x-axis, that would be this angle right here, theta. And this, we know that tan theta is equal to the opposite side divided by the adjacent side, perpendicular divided by base. So that is 8.47 divided by 2. That's really just 8.47 divided by 2. This right here is tan theta. So this is 4.47 and theta, theta would be tan inverse of 4.47. And when you work this out, this comes out to be equal to 76 degrees. So the resultant vector, v plus w is making an angle of 76 with the positive x-axis. Now let's try and check that. Let's try and confirm that using the graphical method of addition, using the head-to-tail method. So to do that, let me make some things disappear. Okay, shifted a couple of things as well. Now to add these two vectors using the graphical method, we can place the tail of the second vector onto the head of the first vector. In a vector, the arrow is the head. This is the tail. So if you're adding v plus w, we can place the tail of w on the head of vector v. So when we do that, when we do that, vector w could look somewhat like this. We have just moved vector w without changing its direction at all. We have just placed its tail on the head of vector v. And the resultant, the addition of v plus w now is when you join the tail of the first vector with the head of the second vector. So that is the tail of the blue vector, this one with the head of the second vector, this one. So when we join this, the resultant looks like this. And we can see how this resultant could be making an angle of 76 degrees with the positive x-axis, pretty much what we got from our calculation. And also it has a very short horizontal component 2i, makes sense, but a huge vertical component 8.47, again, makes sense. So this is how you can add two vectors using the component method. Again, the beauty of this method is that you don't really need to remember any formula. All you need to know is how to resolve a vector into its components.