Video: Vector Addition

Video Transcript

In this lesson, we are going to learn how to add two or more vectors both graphically and using unit vector notation. We will start with the graphical method. Before we begin to learn how to add vectors graphically, let’s refresh our memory on what a vector is and how to represent a vector in graphical form.

A vector is a quantity that has both magnitude and direction. Let’s draw an example of two vectors, 𝐀 and 𝐁. Let’s make vector 𝐀 have a magnitude of four units and be oriented along the horizontal axis towards the right of our screen. When we label vector 𝐀, note that we draw a half arrow over the top of the letter. This is a common convention to show this is a vector. In text form, it is common for the variable to be bolded. We will see this form when we do example problems at the end of our lesson.

We can draw a vector 𝐁 to have the same magnitude of four units. But this time, vector 𝐁 will be oriented vertically towards the top of the screen. Once again, when we label our vector, we will put a half arrow over the top of the letter. Both of these vectors have the same magnitude but are heading in different directions. We need to take extra care and make certain we draw the vectors the correct length and the representation because it will affect the outcome when we add them together.

Before we move on to adding our vectors, let’s draw one last vector that has components along both the horizontal and vertical axes. The graphical representation of vector 𝐂 has a length of six units horizontally and three units vertically. We can see that vector 𝐂 has a greater horizontal magnitude than vector 𝐀 but smaller vertical magnitude when compared to vector 𝐁.

Now that we have done a quick recap on what vectors are and how to draw them graphically, let’s dive into adding vectors, starting by adding them graphically. When we are adding two vectors together, 𝐀 and 𝐁, we can write this as the expression 𝐕 is equal to 𝐀 plus 𝐁, where 𝐕 is the vector formed as a result of summing vectors 𝐀 and 𝐁. The approach we’ll be using to add vectors graphically is the tip-to-tail method.

In the tip-to-tail method, one vector slides over until its tail is on the tip of the other vector. The resultant vector is drawn from the tail of the unmoved vector to the tip of the moved vector. Let’s try out the tip-to-tail method by adding the two vectors 𝐀 and 𝐁 shown on the grid.

Let’s look at the magnitude and direction of our vectors before we add them. Vector 𝐀 has a length of three units horizontally. And vector 𝐁 has a magnitude of two units horizontally and four units vertically. To find the vector that is formed when we add vectors 𝐀 and 𝐁 together, we start by leaving vector 𝐀 where it is on the graph and sliding vector 𝐁 over until the tail of vector 𝐁 is on the tip of vector 𝐀.

It is very important that we keep the size of vector 𝐁’s components horizontally and vertically the same. In this case, two units horizontally and four units vertically. This is the trickiest part to use in the graphical method for vector addition. If we change the components, then our resultant will not be correct.

Now that we have slid vector 𝐁 over, we are ready to draw in our resultant, which goes from the tail of vector 𝐀 all the way to the tip of vector 𝐁. The direction our resultant points in is towards the tip of vector 𝐁 or away from the origin. We can label the resultant as 𝐕 as that is what we called the vector in our expression at the top of the screen.

Adding vectors is not limited to two vectors. Let’s add a third vector 𝐂 to our original problem. Making the resultant vector that has formed 𝐕 be equal to the sum of 𝐀, 𝐁, and 𝐂. We begin again with a grid of the three original vectors that will be added together. Vectors 𝐀 and 𝐁 will be identical to the previous example. But this time, we will also have a vector 𝐂, which has a magnitude of two units along the horizontal to the left of the screen and one unit along the vertical to the bottom of the screen.

We will use the tip-to-tail method to slide both vector 𝐁 and vector 𝐂 over one at a time. The tail of vector 𝐁 slides to the tip of vector 𝐀 just as it did in the previous example. This time, however, we must next slide the tail of vector 𝐂 to the tip of vector 𝐁.

Now, we can draw in our resultant from the origin or the tail of vector 𝐀 to the tip of vector 𝐂. The resultant vector 𝐕 will be pointing away from the origin towards the tip of vector 𝐂. We can see that our resultant vector has components of three units horizontally and three units vertically. This method can continue for as many vectors as we are adding together.

Let’s move on to adding vectors using unit vector notation. We need to recall that a unit vector is a vector of length one. Let’s take a look back at the three vectors we drew at the beginning of our video when we were refreshing our memory on what a vector was and how to draw it graphically.

Let’s start with vector 𝐀. We said that it had a magnitude of four units to the right along the horizontal axis. In unit vector notation, we would say that vector 𝐀 has a value of four 𝐢. The 𝐢 represents the horizontal axis. The expression would look like this. 𝐀 equals four 𝐢, with a little hat over it, where the hat represents that it’s a unit vector. In text form, the 𝐢 would be bolded to show that it’s a unit vector, as we will see in an example problem at the end of this lesson.

Vector 𝐁, we said, had a magnitude of four units to the top of the screen along the vertical axis. In unit vector notation, we would say that vector 𝐁 has a value of four 𝐣, with the expression being 𝐁 equals four 𝐣. And the 𝐣 would have a little hat over it, representing that it’s a unit vector.

Vector 𝐂 had a magnitude of six units to the right along the horizontal axis and three units to the top of the screen along the vertical axis. In unit vector notation, vector 𝐂 would have a value of six 𝐢 plus three 𝐣. The expression would be 𝐂 equals six 𝐢 plus three 𝐣, with both 𝐣 and 𝐢 having hats over them to show that they are unit vectors.

Now that we remember what a unit vector is, let’s go over how to add vectors when they’re in unit vector notation. When adding vectors in unit vector notation, we add the components together. All of the 𝐢-components are added together, and all of the 𝐣-components are added together. The resultant vector 𝐕 in unit vector notation would have an 𝐢-component with all of the individual 𝐢 unit vectors added together and a 𝐣-component with all the individual 𝐣 unit vectors added together.

If we have two vectors in unit vector notation, with vector 𝐀 being four 𝐢 plus nine 𝐣 and vector 𝐁 being equal to seven 𝐢 plus five 𝐣. What would be the sum of vector 𝐀 plus vector 𝐁? With 𝐕 representing the vector that is a resultant of the sum of 𝐀 and 𝐁. We start by adding the 𝐢’s together, four 𝐢 plus seven 𝐢. Four plus seven equals 11. The resultant has a component of 11𝐢. Then we add the 𝐣’s together, nine 𝐣 plus five 𝐣. Nine plus five is 14. The resultant contains a component of 14𝐣. We can say that the resultant vector 𝐕 is 11𝐢 plus 14𝐣.

Even if one of the components is negative, we still add the components. But this time, we account for the negative sign. If vector 𝐁 had a 𝐣-component of negative five instead of positive five, when we added the components together, we wouldn’t get 14𝐣. Instead, we’d get nine plus negative five or four 𝐣.

Let’s try two examples of adding vectors together, one where we add them graphically and one where we add them using unit vector notation.

Which of the vectors 𝐏, 𝐐, 𝐑, 𝐒, or 𝐓 shown in the diagram is equal to 𝐀 plus 𝐁?

The diagram shows seven vectors, including vectors 𝐀 and 𝐁. We are asked, of the other five vectors 𝐏, 𝐐, 𝐑, 𝐒, and 𝐓, which one will be equal to the sum of 𝐀 plus 𝐁. The vectors have been given to us in graphical form. Therefore, we can add the vectors together graphically using the tip-to-tail method.

In the tip-to-tail method, one vector slides over until its tail is on the tip of the other vector. The resultant vector is drawn from the tail of the unmoved vector to the tip of the moved vector. In our problem, we’re gonna slide vector 𝐁 over until the tail of vector 𝐁 is on the tip of vector 𝐀. Making sure that the components of vector 𝐁 as we drew it now are the same as the original vector 𝐁. In this case, that would mean being two units to the right horizontally and three units to the top of the screen vertically.

We draw in our resultant from the tail of vector 𝐀 to the tip of vector 𝐁. The direction our resultant points in is away from the origin or towards the tip of vector 𝐁. We can see if the resultant vector we drew lies along vector 𝐐. Therefore, we can say that vector 𝐐 is the vector that is equal to 𝐀 plus 𝐁. Of the five vectors drawn in the diagram 𝐏, 𝐐, 𝐑, 𝐒, or 𝐓, vector 𝐐 is the one that is equal to 𝐀 plus 𝐁.

Now, let’s look at another example where we add vectors using unit vector notation.

Consider two vectors 𝐀 and 𝐁. 𝐀 is equal to two 𝐢 plus three 𝐣, and 𝐁 is equal to seven 𝐢 plus five 𝐣. Calculate 𝐀 plus 𝐁.

In the problem, 𝐀 and 𝐁 are bolded to show that they’re vectors. Since we’ll be solving this by hand, we’re gonna draw half arrows over our letters to represent that they’re vectors. We align our vectors vertically along with an addition sign so that we may more easily add our components together.

We need to remember that when we’re adding unit vectors, we add the individual components, which means we’ll add the 𝐢’s together and separately we’ll add the 𝐣’s together. We can say that vector 𝐕 will be the resultant vector of 𝐀 plus 𝐁. To determine the components of 𝐕, we start by adding the 𝐢’s together. Two 𝐢 plus seven 𝐢 is nine 𝐢. Then we add the 𝐣’s together. Three 𝐣 plus five 𝐣 is eight 𝐣. When we add two vectors in unit vector notation, with vector 𝐀 being equal to two 𝐢 plus three 𝐣 and vector 𝐁 being equal to seven 𝐢 plus five 𝐣, we get nine 𝐢 plus eight 𝐣.

Key points from our lesson. Add vectors graphically using the tip-to-tail method. When adding vectors using unit vector notation, add the 𝐢-components together and the 𝐣-components together separately.