Lesson Video: Vectors in Terms of Fundamental Unit Vectors Mathematics

Video Transcript

In this video, we’ll learn how to write vectors in component form using fundamental unit vectors.

We know that in two dimensions, there are two components to a vector, called the 𝑥- and 𝑦-components. And that’s where 𝑥 is the horizontal component and 𝑦 is the vertical component. Given the components of a vector in two dimensions, one way we can express it is in component form. For example, if the 𝑥- and 𝑦-components are 𝑎 and 𝑏, respectively, the component form of the vector is 𝑎, 𝑏. And we can represent the vector in the Cartesian plane by an arrow starting at the origin and ending at the point 𝑎, 𝑏, as shown.

But there is another way to express a vector in two dimensions, and that’s in terms of fundamental unit vectors. These are unit vectors denoted 𝐢 and 𝐣 with nonnegative components and length one, where 𝐢 is the unit vector in the horizontal direction and 𝐣 is the unit vector in the vertical direction. 𝐢 has components one, zero, and 𝐣 has components zero, one.

We write 𝐢 and 𝐣 with hats on to distinguish them as the fundamental unit vectors. But you may also see them written with an arrow above or emphasized in bold. Note that the fundamental unit vectors have one nonzero component, which for both vectors is equal to one. From the component form of 𝐢, we can represent it by an arrow starting from a given origin and ending at the point one, zero, which is on the positive 𝑥-axis. Similarly, from the component form of 𝐣, we can represent it in the Cartesian plane as a vertical arrow of length one starting at a given origin. So, 𝐢 and 𝐣 lie parallel to the 𝑥- and 𝑦-axes, respectively, pointing in the positive directions for each respective axis.

Let’s look at an example now of how we might express a vertical vector in terms of the fundamental unit vectors.

Given that the vector 𝐀 has components zero and two, express the vector 𝐀 in terms of the unit vectors 𝐢 and 𝐣.

We’re given a vector in component form, which we need to express in terms of the fundamental unit vectors 𝐢 and 𝐣. To do this, we begin by recalling that we can represent a vector with components 𝑎 and 𝑏 by an arrow starting from the origin and ending at the point with coordinates 𝑎, 𝑏. Since the component form of our vector 𝐀 is zero, two, we represent it by an arrow from the origin to the point zero, two. So how do we express this in terms of our fundamental unit vectors 𝐢 and 𝐣?

Well, recall the component forms of 𝐢 and 𝐣. 𝐢 is equal to one, zero and 𝐣 is equal to zero, one. And they’re represented in the Cartesian plane as shown. In particular, 𝐢 is a horizontal vector of length one, while 𝐣 is a vertical vector also of length one. Since our vector 𝐀 is purely vertical, the 𝑥-component being zero, to express 𝐀 in terms of the fundamental unit vectors, we need only use the vector 𝐣. Since the 𝑦-component of 𝐀 is two, we can achieve this by stacking two copies of 𝐣 on top of each other to produce the vector 𝐀. This tells us that 𝐀 is equal to 𝐣 plus 𝐣, which is two 𝐣. Hence, in terms of the fundamental unit vectors 𝐢 and 𝐣, the vector 𝐀 is equal to two 𝐣.

In this example, we expressed a vertical vector in two dimensions in terms of the fundamental unit vector 𝐣. Now let’s see how to express a vector that’s neither vertical nor horizontal in terms of the unit vectors.

The given figure shows a vector 𝐀 in a plane. Express this vector in terms of the unit vectors 𝐢 and 𝐣.

We’re asked to express the vector shown in the graph in terms of the fundamental unit vectors 𝐢 and 𝐣. So let’s remind ourselves what these look like in the Cartesian plane. 𝐢 is a horizontal vector and 𝐣 is a vertical vector, and they both start at a given origin traveling in the positive 𝑥- and 𝑦-directions, respectively, with length equal to one.

Now, to express the vector 𝐀 in terms of 𝐢 and 𝐣, we need to consider its 𝑥- and 𝑦-components separately. So let’s begin with the 𝑥-component. From the graph, we see that the 𝑥-component is equal to negative three. Using the horizontal unit vector 𝐢, which is going in the positive 𝑥-direction, we flip this unit vector so it becomes negative, as it’s now pointing in the negative 𝑥-direction. And to reach 𝑥 is negative three, we’re adding three copies of negative 𝐢. This tells us that the 𝑥-component of 𝐀 in terms of the fundamental unit vector 𝐢 is negative three 𝐢.

Next, considering the 𝑦-component of 𝐀, we see that this is positive two. And we can produce this by adding two copies of 𝐣. Hence, the 𝑦-component of vector 𝐀 in terms of the fundamental unit vector 𝐣 is two 𝐣.

So now adding the two components together, where by convention we write the 𝑥-component first, we have 𝐀 equals negative three 𝐢 plus two 𝐣.

We can apply the method we used in this example to any vector with integer-valued components. This leads to a general formula for expressing a vector with components 𝑎, 𝑏 in terms of the fundamental unit vectors. That’s 𝑎 times 𝐢 plus 𝑏 times 𝐣. Note that this conversion also works in reverse. Starting from a vector in terms of the fundamental unit vectors, so given 𝑎𝐢 plus 𝑏𝐣, we can covert this to the vector with components 𝑎 and 𝑏.

We’ve shown how to convert between component and unit vector form for integer-valued components of vectors. But actually, this works for any real-valued components. In our next example, we’ll use this to express a two-dimensional vector in terms of fundamental unit vectors.

Express the vector 𝐙 equals negative five over two, negative 19 using the unit vectors 𝐢 and 𝐣.

We’re given a vector in component form, and we want to express this in terms of the fundamental unit vectors 𝐢 and 𝐣. To do this, we recall that a two-dimensional vector with components 𝑎 and 𝑏 can be written in terms of the fundamental unit vectors as 𝑎 times 𝐢 plus 𝑏 times 𝐣. Since our given vector 𝐙 has components 𝑎 equal to negative five over two and 𝑏 equal to negative 19, we can write 𝐙 as negative five over two 𝐢 plus negative 19𝐣. Hence, in terms of the fundamental unit vectors 𝐢 and 𝐣, the vector 𝐙 is equal to negative five over two 𝐢 minus 19𝐣.

In this example, we expressed a two-dimensional vector given in component form in terms of the fundamental unit vectors. Let’s see now how to achieve this when our vector is specified by its beginning and endpoints in the Cartesian plane. One way to write this vector in terms of the fundamental unit vectors would be to first find the coordinate form of the vector and then convert the form. But we don’t have to go through the coordinate form to achieve this. Instead, we can just identify the horizontal and vertical components of this vector so that we can write it as the sum of horizontal and vertical vectors. Let’s see how this works in an example.

The given figure shows a vector in a plane. Express this vector in terms of the unit vectors 𝐢 and 𝐣.

We’re asked to write the vector shown in the graph in terms of the fundamental unit vectors 𝐢 and 𝐣. And we recall that the unit vectors are defined by 𝐢 equals the vector with components one, zero and 𝐣 is the vector with components zero, one. In other words, these are unit vectors pointing in the positive horizontal and vertical directions of the 𝑥- and 𝑦-axes, respectively. These are the fundamental unit vectors.

Note that unit vectors need not necessarily start at the origin. They describe moving a distance of one in either the horizontal or vertical direction from a given initial point. So we need to express the given vector as a sum of horizontal and vertical unit vectors. Let’s first identify the relevant horizontal and vertical vectors on our graph.

First, our initial point is the point two, negative two. And starting from here, we see that the horizontal vector spans two grid lengths in the positive 𝑥-direction. So the vector’s horizontal component is equal to positive two, which we can write as the vector with components two, zero. And this can be written as two times the vector one, zero, which is two times the fundamental unit vector 𝐢. Similarly, the vertical vector spans 10 grid lengths and points in the positive 𝑦-direction from our initial point so that its vertical component is positive 10. This gives us that the vertical vector is 10 times the fundamental unit vector 𝐣. Adding these two vectors together will produce the given vector. Hence, the given vector is equal to two 𝐢 plus 10𝐣.

In this example, we expressed a graphed vector on a grid in terms of the fundamental unit vectors. This is a perfectly valid method. But to use it, we need to graph the points on the coordinate plane. It’s useful to know the formula for achieving this when we only have the coordinates of the initial point and the endpoint of the vector.

Let’s consider a vector from initial point 𝐴 with coordinates 𝑥 one, 𝑦 one to endpoint 𝐵 with coordinates 𝑥 two, 𝑦 two. In this case, the 𝑥-component of the vector 𝐀𝐁 is 𝑥 two minus 𝑥 one and the 𝑦-component is 𝑦 two minus 𝑦 one. We can then write the vector 𝐀𝐁 from initial point 𝐴 to endpoint 𝐵 in terms of the fundamental unit vectors as 𝑥 two minus 𝑥 one 𝐢 plus 𝑦 two minus 𝑦 one 𝐣.

It’s worth noting here that it’s very important we’re clear which is the initial and which is the endpoint of a vector. Here, our initial point is 𝐴 and the endpoint is 𝐵. However, if we start from 𝐵 and go in the opposite direction from 𝐵 to 𝐴, the vector 𝐁𝐀 has components 𝑥 one minus 𝑥 two and 𝑦 one minus 𝑦 two, which are not the same as those of vector 𝐀𝐁 in the opposite direction.

In our final example, let’s apply this formula to write a two-dimensional vector in terms of the fundamental unit vectors 𝐢 and 𝐣.

Given that 𝐴 has coordinates two, three and 𝐵 has coordinates five, nine, express the vector 𝐀𝐁 in terms of the unit vectors 𝐢 and 𝐣.

In this example, given the initial and endpoint of a vector in two dimensions, we want to express the vector in terms of the fundamental unit vectors 𝐢 and 𝐣. To do this, we recall that a vector 𝐀𝐁 with initial point 𝐴 𝑥 one, 𝑦 one and endpoint 𝐵 𝑥 two, 𝑦 two can be written in terms of the fundamental unit vectors as 𝐀𝐁 is equal to 𝑥 two minus 𝑥 one times 𝐢 plus 𝑦 two minus 𝑦 one times 𝐣.

We’re given the points 𝐴 and 𝐵. And since 𝐴 is the initial point and 𝐵 the endpoint, we have 𝑥 one equals two and 𝑥 two equals five, 𝑦 one equals three and 𝑦 two equals nine. Using the formula gives us 𝐀𝐁 is equal to five minus two times 𝐢 plus nine minus three times 𝐣, that is, three 𝐢 plus six 𝐣. Hence, in terms of the fundamental unit vectors, the vector with initial point 𝐴 two, three and endpoint 𝐵 five, nine is 𝐀𝐁 equals three 𝐢 plus six 𝐣.

Let’s now complete this video by recapping a few important points we’ve covered. The fundamental unit vectors 𝐢 and 𝐣 are horizontal and vertical vectors of length one pointing in the positive 𝑥- and 𝑦-directions, respectively. The component form of 𝐢 is the vector one, zero, and the component form of 𝐣 is the vector zero, one. A vector in component form 𝑎, 𝑏 can be written in terms of unit vectors 𝐢 and 𝐣 as 𝑎 times 𝐢 plus 𝑏 times 𝐣. Similarly, given a vector in terms of the fundamental unit vectors, we can write this in component form 𝑎, 𝑏. And finally, a vector 𝐀𝐁 with initial point 𝐴 𝑥 one, 𝑦 one and endpoint 𝐵 𝑥 two, 𝑦 two can be written in terms of the fundamental unit vectors as 𝐀𝐁 equals 𝑥 two minus 𝑥 one times 𝐢 plus 𝑦 two minus 𝑦 one times 𝐣.