In this video, we’re talking about finding the components of a vector. We’re going to learn how to find these components graphically using a grid as well as using trigonometry. Now, as we get started, let’s point out that all vectors are made up of components. We could say that components are the pieces we add together to get a vector.
Given a vector, say that we start with this vector we’ve called 𝐕, we can see its components by drawing it on a grid. First, we sketch in an 𝑥𝑦-coordinate frame and then add in grid lines. We can see that our vector extends one, two, three grid spaces along this 𝑥-axis. We would say then that the vector’s 𝑥-component, we can call it 𝑉 sub 𝑥, equals three. The 𝑦-component of 𝐕 goes one, two, three, four, five, six units up the 𝑦-axis. So we can say that 𝑉 sub 𝑦 is six. These are the components of the vector 𝐕.
Now, in general, if we have a two-dimensional vector, we can call that vector 𝐀, then we can write the vector 𝐀 like this. Here, 𝐴 sub 𝑥 and 𝐴 sub 𝑦 are its 𝑥- and 𝑦-components. And notice that we don’t simply add these components together to get the vector 𝐀. First, each one has to be multiplied by the correct unit vector. In the case of 𝐴 sub 𝑥, this is the 𝐢 hat unit vector here. And for 𝐴 sub 𝑦, it’s the 𝐣 hat unit vector.
Recall that a unit vector is a vector with a magnitude or a length of one. If we drew in our 𝐢 hat unit vector on our grid, it would look like this, while the 𝐣 hat unit vector would look like this. These vectors are important because by themselves the 𝑥- and 𝑦-components of a given vector are scalar quantities. So they can’t give us a vector all by themselves. That is, they have a magnitude or a length, but they don’t have any direction. That’s where the unit vectors come in.
This first term on the right in our equation says that the vector 𝐀 has a length 𝐴 sub 𝑥, and it’s in this particular direction, the 𝑥-direction. Likewise, it extends a length 𝐴 sub 𝑦 in the 𝑦-direction. All this to say, the components of a vector are not themselves vectors. Actually, they’re scalar quantities. We saw that here with our vector 𝐕, where we found out its 𝑥-component is three, not a vector but a scalar, and its 𝑦-component is a scalar too.
So to write out the vector 𝐕 in terms of its components, we would say that it has a length of 𝑉 sub 𝑥, its 𝑥-component, in the 𝐢 hat direction. And then we add to that another vector, its 𝑦-component 𝑉 sub 𝑦 times the unit vector 𝐣 hat. Filling in for our known values, we could write this as three 𝐢 plus six 𝐣. And this is called the component form of our vector 𝐕. It’s expressed in terms of a horizontal or 𝑥-component and a vertical or 𝑦-component.
Now, we mentioned earlier that there’s more than one way to solve for the components of a vector. In the case of vector 𝐕, we’ve used grid spacings to calculate these components. But let’s imagine that instead we’re given a vector, we’ll call it vector 𝐑, on an unmarked 𝑥𝑦-plane. In this case, we’re not given any grid marks to work off of. But say that we are given the magnitude of our vector as well as the angle that it makes with the horizontal. It’s possible to solve for the components of 𝐑 using just this information.
Let’s remember that any two-dimensional vector has a horizontal as well as a vertical component. If we were to sketch in that horizontal component, it would look like this and the vertical component like this, which now that we think about it is equal in length to this dashed line. Since the vertical and horizontal components of any vector are perpendicular to one another, we know that this angle is a right angle. And now we have a right triangle where this side is the hypotenuse and these lengths are the other two sides. And notice this: the shorter two sides in this triangle are the 𝑥- and 𝑦-components of our vector 𝐑.
Whenever we have a right triangle and one of the other interior angles is also known, there are specific trigonometric relationships between the lengths of the three sides of this triangle. In a right triangle, the side opposite the right angle is always the hypotenuse. Then, because this angle, we’ve called it 𝜃, is the other one we know, we call this side of the triangle the opposite side, that is, opposite from 𝜃, and this the adjacent side, because it’s adjacent or next to 𝜃. Connecting this triangle to the one created by our vector 𝐑, notice that the adjacent and opposite sides of the known angle of 37 degrees are what we want to solve for. Those are the horizontal and vertical components of 𝐑, respectively.
Now, these different sides of our right triangle relate to one another through trigonometric functions. Here’s what we mean by that. If we take the sin of the angle 𝜃, then that’s equal to the length of the side opposite the angle 𝜃 divided by the hypotenuse length. If we multiply both sides of this equation by the length of the hypotenuse, then we end up with an equation where the opposite side length is the subject. In other words, the hypotenuse length times the sin of 𝜃 will give us the vertical component of our vector.
When it comes to the vector 𝐑, we see that its hypotenuse is nine and 𝜃 is 37 degrees. So if we multiply nine by the sin of 37 degrees, then we’ll get the vertical component of 𝐑. To two significant figures, this is 5.4. When it comes to solving for the horizontal component of 𝐑, we can note that the cos of the angle 𝜃 is equal to the adjacent side length over the hypotenuse. Rearranging, this gives us that the hypotenuse of our triangle times the cos of 𝜃 equals that adjacent side length. And that in the case of our vector 𝐑 is the horizontal component. 𝑅 sub 𝑥 is equal to nine times the cos of 37 degrees, which rounded to two significant figures is 7.2. We’ve solved then for the vertical and horizontal components of 𝐑. So we could write it out in component form like this. We say then that vector 𝐑 has a magnitude of nine, a direction of 37 degrees above the horizontal, a horizontal component of 7.2, and a vertical component of 5.4.
Knowing all this about the components of vectors, let’s get a bit of practice through an example.
The vector 𝐀 can be written in the form 𝑎 𝑥 times 𝐢 hat plus 𝑎 𝑦 times 𝐣 hat. What is the value of 𝑎 𝑥? What is the value of 𝑎 𝑦?
Alright, in this description of the vector 𝐀, 𝑎 𝑥 and 𝑎 𝑦 are its 𝑥- and 𝑦-components. These are scalar quantities that show the length of vector 𝐀 in the horizontal and vertical directions. If we look at this sketch of vector 𝐀 with the grid around it, we see that there’s a horizontal axis, we’ll call that the 𝑥-axis, and a vertical one we’ll call 𝑦.
So looking at this first part of our question, 𝑎 sub 𝑥 will be the amount that vector 𝐀 lies along this 𝑥-axis. This is called the horizontal component of 𝐀. And we find its value by projecting 𝐀 downward perpendicularly onto this 𝑥-axis. On this graph then, 𝑎 sub 𝑥 would look like this. It’s a length that covers one, two, three, four, five, six, seven grid spaces. And so that’s the value of 𝑎 sub 𝑥. Knowing this, we then want to figure out what the 𝑦-component of our vector 𝐀 is. This time, we’ll project our vector perpendicularly onto the vertical axis. 𝑎 sub 𝑦 then would look like this. We can count squares to figure out the length of this line. It’s one, two, three, four units long. So 𝑎 sub 𝑦 is four. We’ve solved then for both the horizontal as well as the vertical components of vector 𝐀.
Let’s look now at another example.
Write 𝐀 in component form.
Here, we see this vector 𝐀 drawn on a grid. And we can see the vector starts at the origin of a coordinate frame. Let’s call the horizontal axis the 𝑥-axis and the vertical one the 𝑦. Now, when we go to write this vector 𝐀 in component form, that means we’ll write it in terms of an 𝑥- and a 𝑦-component, also called a horizontal and vertical component. If we call the 𝑥-component of vector 𝐀 𝐴 sub 𝑥 and the 𝑦-component 𝐴 sub 𝑦, then we can multiply each one of these components by the appropriate unit vector. The unit vector for the 𝑥- or horizontal direction is 𝐢 hat, and the unit vector for the vertical or 𝑦-direction is 𝐣 hat. By themselves, the 𝑥- and 𝑦-components of vector 𝐀 are not vectors; they’re scalar quantities. But when we multiply these scalars by a vector, the unit vectors, the result is a vector.
Finally, adding these vector components together, we’ll get the vector 𝐀. Expressing 𝐀 this way is known as writing it in component form. So then, what are 𝐴 sub 𝑥 and 𝐴 sub 𝑦? To figure that out, we’ll need to look at our grid. Starting with 𝐴 sub 𝑥, that’s equal to the horizontal component of this vector 𝐀. In other words, if we project this vector perpendicularly onto the 𝑥-axis, then the length of that line segment, this length here, is 𝐴 sub 𝑥. In terms of the units of this grid, that length is one, two, three units long. And notice that we moved to the left of the origin, that is, into negative 𝑥-values. So even though this horizontal orange line is three units long, we say that the 𝑥-component of 𝐀 is negative three. This is because the projection of vector 𝐀 onto the horizontal axis goes negative three units in the 𝑥-direction.
To find the vertical component of 𝐀, we’ll follow a similar process. Once again, we project vector 𝐀 perpendicularly, this time onto the vertical axis. And it’s the length of this line that tells us the vertical or 𝑦-component of 𝐀. We see that this is one, two units long and that this is in the positive 𝑦-direction. 𝐴 sub 𝑦 then is equal to positive two. And now we can write out 𝐀 in its component form. Vector 𝐀 is equal to negative three times the 𝐢 hat unit vector plus two times the 𝐣 hat unit vector.
Let’s look now at one last example exercise.
The diagram shows a vector 𝐀 that has a magnitude of 22. The angle between the vector and the 𝑥-axis is 36 degrees. Work out the horizontal component of the vector. Give your answer to two significant figures.
Alright, we see this vector 𝐀 sketched out. And we’re told it has a length or magnitude of 22. Along with this, we know that the vector forms an angle of 36 degrees with the positive 𝑥-axis. Our goal is to solve for its horizontal component. This is equal to the horizontal projection of our vector onto this axis.
As we go about solving for the length of this orange line, let’s note that our dashed line intersects our horizontal axis at a right angle. In other words, we have here a right triangle. Here’s the hypotenuse, here’s another side, and here’s the third. In solving then for the horizontal component of our vector, we’re solving for one of the sides of this right triangle. We can do this using trigonometry.
Let’s remember that, given a right triangle, if we know one of the other interior angles, then we can define the sides of this right triangle as hypotenuse ℎ, the side opposite our angle 𝜃 𝑜, and the side adjacent to that angle 𝑎. Set up this way, it’s the adjacent side, what we’ve called 𝑎 over here, that we want to solve for to figure out our horizontal component.
Now, if we were to take the cos of this angle 𝜃, then that would equal the ratio of our adjacent side length to our hypotenuse. Or multiplying both sides of this equation by the hypotenuse, canceling that factor out on the right, we have that the adjacent side of our right triangle equals the cos of 𝜃 times ℎ. This relates to our situation with vector 𝐀 because in this case we know the length of our hypotenuse and we also know this angle. So we can actually say that the length of our triangle’s hypotenuse, 22, multiplied by the cos of our angle of 36 degrees is equal to what we’ll call 𝐴 sub 𝑥, the horizontal component of the vector 𝐀. When we enter this expression on our calculator and keep two significant figures, our answer is 18. This is the horizontal component of the vector 𝐀.
Let’s finish up our lesson now by reviewing a few key points. In this lesson, we saw that a vector, say we have a vector 𝐀, can be written in terms of components 𝐴 𝑥 and 𝐴 𝑦. Here, 𝐴 𝑥 represents the horizontal component of our vector and 𝐴 𝑦 the vertical component. We also learn that a vector’s components are scalar values that can be determined in one of two ways: first from a grid or second using trigonometry. And lastly, considering the trigonometry of a right triangle, we saw that, given an interior angle 𝜃, which is different from the right angle in the triangle, the sin of 𝜃 is equal to the ratio of the opposite side length to the hypotenuse, while the cos of 𝜃 is equal to the adjacent side length to the hypotenuse. Here, the opposite and adjacent side lengths often represent the vertical and horizontal components, respectively, of a vector.