Lesson Video: Vector Subtraction Physics

In this video, we will learn how to subtract one vector from another in two dimensions, using both graphical and algebraic methods.

13:34

Video Transcript

In this lesson, we’re going to learn how to subtract one vector from another vector, both graphically and using unit vector notation.

We will start with the graphical method. Before we begin to learn how to subtract vectors graphically, let’s refresh our memory on what a vector is and how to represent a vector graphically. 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 five units and be oriented horizontally to the right of our screen. Note that when we label our vector 𝐀, we draw a half arrow over the top of the letter. This is a common convention to show that a variable 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 vector 𝐁 to have the same magnitude of five units, but we can orient it vertically to the top of our screen. Both of these vectors have the same magnitude but are heading in different directions. We need to take extra care and make sure that we draw the vectors to the correct length in the representation because it will affect the outcome when we subtract them. Before we move on to subtracting 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 seven units along the horizontal to the right of our screen and four units along the vertical to the top of our screen. We can see that vector 𝐂 has a greater horizontal magnitude than vector 𝐀 but smaller vertical magnitude than vector 𝐁. Now that we have done a quick recap on what vectors are and how to draw them graphically, let’s dive into how to subtract vectors graphically.

When we’re subtracting vector 𝐁 from vector 𝐀, we can write this expression as 𝐕, the resultant vector, is equal to vector 𝐀 minus vector 𝐁. The approach we’ll be using to subtract vectors graphically is to add the negative of vector 𝐁 to vector 𝐀 and then apply the tip-to-tail method of adding vectors. We can do this because subtracting vector 𝐁 from vector 𝐀 is the same thing as adding negative vector 𝐁 to vector 𝐀.

When we talk about the negative of a vector, in this case, negative vector 𝐁, that means we need to rotate our vector 180 degrees from its starting position. Our new vector will be pointing in the complete opposite direction both horizontally and vertically from where the vector was originally. In the tip-to-tail method, one vector slides over until its tail is on the tip of the other vector. The resultant is drawn from the tail of the unmoved vector to the tip of the moved vector. Let’s subtract vector 𝐁 from vector 𝐀, which are both shown on the grid.

Before we subtract our vectors, let’s take a look at both their magnitudes and directions. Vector 𝐀 has a length of seven units along the horizontal axis to the right of our screen, and vector 𝐁 has a magnitude of three units along the horizontal axis to the right of our screen and four units vertically to the top of our screen. To Find the vector that is formed when we subtract vector 𝐁 from vector 𝐀, we start by drawing a vector that represents the negative of vector 𝐁. To do this, we rotate vector 𝐁 180 degrees until it’s pointing in the opposite direction both horizontally and vertically from where it was originally drawn. The length of the vector will still be the same, even though the orientation has changed.

For our example, vector 𝐁 was three units to the right and four units up. Therefore, vector negative 𝐁 would be three units to the left and four units down. Another way to draw the negative vector is to simply flip the tip and the tail of the original vector. In this case, vector negative 𝐁 would have its tail on the tip of vector 𝐁 and the tip of vector negative 𝐁 would be on the tail of vector 𝐁. It doesn’t matter which method you use as they both result in the negative of vector 𝐁.

For this example, we’ll use vector negative 𝐁 that’s shown by the dotted yellow line. We can now apply the tip-to-tail method. We leave vector 𝐀 where it is and slide vector negative 𝐁 over until the tail of vector negative 𝐁 is on the tip of vector 𝐀. It is important to slide over vector negative 𝐁 and not vector 𝐁 because otherwise we would be adding our original vectors together rather than subtracting them. Our resultant is drawn from the tail of vector 𝐀 to the tip of vector negative 𝐁. The direction our resultant points in is away from the origin or towards the tip of vector negative 𝐁. We can label the resultant vector 𝐕 as that is what we called our vector at the top of the screen. Our resultant vector has a length of four units horizontally to the right and four units vertically to the bottom of the screen.

Another way to subtract vectors is to use unit vector notation. Let’s recall that a unit vector is a vector of length one. Let’s take a look back at the three vectors that 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 vector 𝐀 had a magnitude of five units along the horizontal axis to the right of our screen. In unit vector notation, we would say that vector 𝐀 has a value of five 𝐒 where the vector 𝐒 points in the horizontal direction. The expression would be vector 𝐀 is equal to five 𝐒, where the 𝐒 has a little hat to represent that it’s a unit vector. If this was in text form, our 𝐒 could be bolded to show that it’s a unit vector.

Vector 𝐁 also had a magnitude of five units, except this time, it was along the vertical axis. In unit vector notation, we would say that vector 𝐁 has a value of five 𝐣, where the vector 𝐣 points in the vertical direction. The expression is vector 𝐁 is equal to five 𝐣, with the 𝐣 having a little hat over it to represent its unit vector notation. Vector 𝐂 had a magnitude of seven units to the right along the horizontal axis and four units to the top of the screen along the vertical axis. In unit vector notation, vector 𝐂 would have a value of seven 𝐒 plus four 𝐣, where the vector 𝐒 points in the horizontal direction and the vector 𝐣 points in the vertical direction.

Now that we’ve done a quick recap on unit vectors, let’s go over how to subtract vectors when they’re in unit vector notation. When we’re subtracting vectors that are in unit vector notation, we need to subtract the individual components. This means we have to subtract the 𝐒 hats from each other and the 𝐣 hats from each other separately. The resultant vector 𝐕 would have an 𝐒-component that would be equal to the subtraction of the individual 𝐒-components and the 𝐣-component that would be equal to the subtraction of the individual 𝐣-components.

If we have two vectors in unit vector notation with vector 𝐀 being equal to eight 𝐒 plus 10 𝐣 and vector 𝐁 being equal to three 𝐒 plus two 𝐣, what would be the value of the resultant vector that’s equal to vector 𝐀 minus vector 𝐁? Vector 𝐕 will be the resultant vector of vector 𝐀 minus vector 𝐁. We will start by subtracting the 𝐒-components. Eight 𝐒 minus three 𝐒 is five 𝐒. Then, we subtract the 𝐣-components. 10𝐣 minus two 𝐣 is eight 𝐣. Our resultant vector 𝐕 has a value of five 𝐒 plus eight 𝐣.

Now, let’s look what happens when we switch the order of subtraction, meaning that this time, our resultant vector 𝐕 will be equal to vector 𝐁 minus vector 𝐀. Let’s look at it both graphically and in unit vector notation. We’ll use the same vectors that we did when we were subtracting graphically. We are subtracting vector 𝐀 from vector 𝐁. So let’s start by writing vector 𝐁 in unit vector notation. Vector 𝐁 has a horizontal component of three units to the right of the screen. Therefore, the 𝐒-component of vector 𝐁 will be three. Vector 𝐁 also has a vertical component of four units to the top of the screen. So, the 𝐣-component of vector 𝐁 will be four. Vector 𝐀 has a value of seven units to the right horizontally. In unit vector notation, we would say that vector 𝐀 is equal to seven 𝐒 plus zero 𝐣.

Let’s subtract our vectors graphically. Remember that if we’re subtracting vector 𝐀, that’s the same thing as adding negative vector 𝐀. Let’s remember from the beginning of the video that drawing in the negative of a vector is the same thing as drawing in a vector that’s 180 degrees to the original. Since vector 𝐀 has a value of seven units to the right horizontally, vector negative 𝐀 will have a value of seven units to the left horizontally. Then, we apply the tip-to-tail method by sliding vector negative 𝐀 over until the tail of vector negative 𝐀 is on the tip of vector 𝐁. Our resultant vector is drawn from the tail of vector 𝐁 to the tip of vector negative 𝐀 and points away from the origin or towards the tip of vector negative 𝐀.

We can see that our resultant vector 𝐕 has a length of four units along the horizontal to the left of our screen and a length of four units along the vertical to the top of the screen. In unit vector notation, our resultant vector 𝐕 would be equal to negative four 𝐒 for the four units to the left of our screen along the horizontal plus four 𝐣 for the four units along the vertical to the top of our screen. We can verify our resultant vector by subtracting vector 𝐀 from vector 𝐁, using unit vector notation.

Three 𝐒 minus seven 𝐒 is equal to negative four 𝐒 and four 𝐣 minus zero 𝐣 is equal to four 𝐣. This matches the results that we’ve got for the graphical method. Now let’s compare the resultant of vector 𝐁 minus vector 𝐀 to our earlier resultant of vector 𝐀 minus vector 𝐁. We have drawn a pink dotted vector to represent the resultant vector of 𝐀 minus 𝐁. Notice that the resultant vector of 𝐀 minus 𝐁 is the negative of the vector that represents 𝐁 minus 𝐀. If we were to write this as an expression, we could say that vector 𝐀 minus vector 𝐁 is equal to the negative of vector 𝐁 minus vector 𝐀.

Comparing the resultant vectors for vector 𝐀 minus vector 𝐁, which is equal to four 𝐒 minus four 𝐣, and vector 𝐁 minus vector 𝐀, which is equal to negative four 𝐒 plus four 𝐣. We see once again that vector 𝐀 minus vector 𝐁 is equal to the negative of vector 𝐁 minus vector 𝐀. Now, let’s take a look at two examples of subtracting vectors, one where we subtract the vectors graphically and one where we subtract them using unit vector notation.

The diagram shows seven vectors 𝐀, 𝐁, 𝐏, 𝐐, 𝐑, 𝐒, and 𝐓. Which of the vectors is equal to 𝐀 minus 𝐁?

We are being asked to find the vector resultant of vector 𝐀 minus vector 𝐁. Since our vectors have been given to us graphically, we can solve for the vector subtraction using a graphical method. We need to remember that when we’re subtracting vector 𝐁 from vector 𝐀, it’s the same thing as adding the negative of vector 𝐁 to vector 𝐀. To solve our problem, we are going to need to draw in a vector that’s the negative of vector 𝐁. A negative vector is a vector that is rotated 180 degrees from the original vector. Vector 𝐁 had a magnitude of three units to the right of the screen. Therefore, vector negative 𝐁 will have a value of three units to the left of the screen. Vector 𝐁 also had a length of five units to the top of our screen and therefore vector negative 𝐁 will have a value of five units to the bottom of the screen.

Now that we have drawn in vector negative 𝐁, we can add vector 𝐀 and vector negative 𝐁 together 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 is drawn from the tail of the unmoved vector to the tip of the moved vector. For our problem, we’re gonna slide vector negative 𝐁 over until it’s on the tip of vector 𝐀. Then, we draw in a resultant vector from the tail of vector 𝐀 to the tip of vector negative 𝐁. The resultant points away from the origin towards the tip of vector negative 𝐁. We can see that vector 𝐐 overlaps our resultant for vector 𝐀 minus vector 𝐁. Therefore, we can say that of the seven vectors shown vector 𝐐 is equal to vector 𝐀 minus vector 𝐁.

Now, let’s subtract two vectors using unit vector notation.

Consider the two vectors 𝐀 and 𝐁, where 𝐀 is equal to eight 𝐒 plus 10𝐣 and 𝐁 is equal to three 𝐒 plus two 𝐣. Calculate 𝐀 minus 𝐁.

In our problem, we are given vector 𝐀 and 𝐁 in unit vector notation. Therefore, we need to remember that when we subtract vectors in unit vector notation, we need to subtract the individual components. This means that we’ll subtract the 𝐒’s and separately we’ll subtract the 𝐣’s. We can align the vectors vertically along with the subtraction sign to make the subtraction of the components easier. We put hats over our 𝐒 and 𝐣 unit vectors as we are writing our workout by hand. But in the text form of the problem, 𝐒 and 𝐣 were bolded to show that they were unit vectors.

We can use the vector 𝐕 to represent the resultant vector of 𝐀 minus 𝐁. To find our resultant, we start with the 𝐒-components. Eight 𝐒 minus three 𝐒 is five 𝐒. Then, we subtract the 𝐣’s. 10𝐣 minus two 𝐣 is eight 𝐣. When we subtract two vectors 𝐀 and 𝐁, where 𝐀 is equal to eight 𝐒 plus 10𝐣 and 𝐁 is equal to three 𝐒 plus two 𝐣, we get a resultant of five 𝐒 plus eight 𝐣.

Key Points from the Lesson

When subtracting vectors graphically, draw the negative vector for the vector being subtracted and then apply the tip-to-tail method. When subtracting vectors in unit vector notation, subtract the individual 𝐒-components and the individual 𝐣-components separately. Vector 𝐀 minus vector 𝐁 is equal to the negative of vector 𝐁 minus vector 𝐀.

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