Explainer: Components of a Vector

In this explainer, we will learn how to find the components of a given two-dimensional vector.

Vectors can be represented in a suitable space by a line segment with a specific length, or magnitude, and direction. This means that we can think of vectors as defining movement: traveling from a starting point in a given direction for a specified distance. Vectors can also represent, for example, force or acceleration.

We can also define a two-dimensional vector using its horizontal and vertical components by writing it either in the form 𝑎,𝑏 or in the form 𝑎+𝑏ij. Here, we will discuss these horizontal and vertical components and introduce the i and j unit vector notation.

Any two-dimensional vector 𝑎,𝑏 has two components. When a vector is represented graphically by a line segment on a graph, 𝑎 represents the horizontal difference between the 𝑥-coordinates of its endpoints, and 𝑏 represents the vertical difference between the 𝑦-coordinates. The numbers 𝑎 and 𝑏 can also be thought of as representing movement in the horizontal and vertical directions.

Example 1: Finding the Horizontal and Vertical Components of a Vector

Consider the vector 𝐴𝐵 for 𝐴(3,2) and 𝐵(6,9). Write 𝐴𝐵 in the form 𝑎,𝑏.

Answer

To find the horizontal and vertical components of 𝐴𝐵, that is, to find 𝑎 and 𝑏 such that 𝐴𝐵=𝑎,𝑏, we can consider the horizontal and vertical distances from A to B. Another way to look at this is to find how far we have to move in the horizontal direction (𝑎) and the vertical direction (𝑏) to get from A to B.

The horizontal distance is equal to the difference between the 𝑥-coordinates; this is 63=3. The vertical distance is equal to the difference between the 𝑦-coordinates, which is 92=7.

Hence, we can write the vector from A to B as 𝐴𝐵=3,7.

How to: Write a Vector in Component Form

To write a vector 𝐴𝐵 where 𝐴(𝑥,𝑦) and 𝐵(𝑥,𝑦) in the form 𝑎,𝑏, first calculate the difference between the 𝑥-coordinates to obtain 𝑎 and then calculate the difference between the 𝑦-coordinates to obtain 𝑏: 𝐴𝐵=𝑎,𝑏,𝑎=𝑥𝑥𝑏=𝑦𝑦.whereand

In this form, 𝑎 represents the 𝑥-component, or the horizontal distance, and 𝑏 represents the 𝑦-component, or the vertical distance.

This notation can be extended to any number of dimensions. If we had a three-dimensional coordinate system with 𝑥-, 𝑦-, and 𝑧-coordinates, then we would add a third number to represent the 𝑧-component: 𝑎,𝑏,𝑐.

Example 2: Finding the Horizontal and Vertical Components of a Vector

Consider the vector in the diagram.

  1. What are the coordinates of its terminal point?
  2. What are the coordinates of its initial point?
  3. What are the components of the vector?

Answer

Part 1

When a vector is represented on a plane, the terminal point of the vector is the point where the head of the arrow is. We can think of this as where the vector is pointing. From the diagram, we can see that (7,1) is the point where the arrow ends.

Part 2

Similarly, the initial point is the point from which the arrow starts. Hence, by looking at the diagram, we can see that (1,2) is the initial point.

Part 3

The first component of the vector can be found by considering the difference between the 𝑥-coordinates of the terminal and initial points; that is, the first component (or, equivalently, the 𝑥-component) of the vector v is 7(1)=6. As for the second component, it is the difference in the 𝑦-coordinates of the terminal and initial points. Hence, the second component (or, equivalently, the 𝑦-component) is given by 12=3. When we write vectors in component form, we use the notation 6,3.

Next, we define two special vectors which have length 1. The vector i represents moving a distance of positive one in the 𝑥-direction, and the vector j represents moving a distance of positive one in the 𝑦-direction.

Definition: The Unit Vectors

Using the notation we described earlier, we can define ij=1,0=0,1,and where 1,0 represents moving one unit in the 𝑥-direction and no units in the 𝑦-direction and 0,1 represents moving no units in the 𝑥-direction and one unit in the 𝑦-direction.

It is important to note that these special vectors do not have to start at the origin. These unit vectors just describe moving a distance of one in either the horizontal or the vertical direction; the start point can change.

For example, as seen in the figure below, the unit vector i can represent moving from (2.5,2) to (3.5,2) since the 𝑥-coordinate increases by one and the 𝑦-coordinate does not change.

The unit vector j can represent moving from (4,3) to (4,4) since the 𝑥-coordinate does not change and the 𝑦-coordinate increases by one.

Now, we will investigate how to add and subtract with these unit vectors.

The vector 𝑎i represents moving along 𝑎 copies of i. We can write 𝑎=𝑎,0i which means that we move 𝑎 units in the horizontal direction and no units in the vertical direction.

The vector 𝑏j represents moving along 𝑏 copies of j. We can write 𝑏=0,𝑏j which means that we move 𝑏 units in the vertical direction and no units in the horizontal direction.

For example, shown in orange in the figure below is the vector 2=2,0i.

Once we have vectors of the form 𝑎i and 𝑏j we can add them together to describe any vector in the form 𝑎+𝑏ij.

Example 3: Writing Vectors as the Sum of Unit Vectors

Using that each square in the grid has length 1, write the vector 𝐴𝐵 in the form 𝑎+𝑏ij and then in the form 𝑎,𝑏.

Answer

From the start point 𝐴, we move +2 units in the horizontal direction (which represents the vector 2i) and then move +3 units in the vertical direction (which represents the vector 3j) to get to the point 𝐵.

The vector 𝐴𝐵, which represents traveling directly from 𝐴 to 𝐵, is then the sum of these unit vectors.

Hence, 𝐴𝐵=2+3=2,3.ij

How to: Write a Vector as the Sum of Unit Vectors

In general, if we travel from a starting point 𝐶(𝑥,𝑦) to a finishing point 𝐷(𝑥,𝑦) then this describes a vector 𝐶𝐷 which represents traveling a distance of (𝑥𝑥) in the 𝑥-direction and then a distance of 𝑦𝑦 in the 𝑦-direction.

We can write this vector in two ways: 𝐶𝐷=𝑥𝑥,𝑦𝑦 or 𝐶𝐷=(𝑥𝑥)+(𝑦𝑦).ij

Using the vectors i and j allows us to describe the vector in terms of the number of horizontal and vertical steps of length one we would have to take to get from the start point to the end point.

Note that negative coefficients of i and j represent moving in the opposite direction.

For example, the vector 𝐴𝐵=2,3, which represents traveling two negative units in the 𝑥-direction and three negative units in the 𝑦-direction, as seen in the figure above, can be written as (2)+(3)ij or 𝐴𝐵=23.ij

Sometimes, we have to solve word problems with vectors where the main skill is interpreting the wording of the question and translating it into mathematics.

Example 4: Word Problems Involving Components of Vectors

A body moved 190 cm due east, where i and j are two unit vectors in the east and north directions respectively. Express its displacement in terms of the two unit vectors i and j.

Answer

We have been told that i represents the easterly direction and j the northerly direction. The body moved east. Therefore, the vector expressing this displacement will have no northerly component. Therefore, the coefficient of j will be zero. We know that it moved 190 cmin the easterly direction; therefore, the coefficient of i will be 190. Therefore, the displacement of the body can be written as 190+0=190.ijicm

Example 5: Adding Vectors Given as Sums of Unit Vectors

Let 𝐴𝐵=3+4ij and 𝑋𝑌=2+3ij. Find 𝐴𝐵+𝑋𝑌.

Answer

When we add vectors, we add the 𝑥-components together (to calculate the total horizontal movement) and then add the 𝑦-components together (to calculate the total vertical movement). So, 𝐴𝐵+𝑋𝑌=(3+4)+(2+3)=(32)+(4+3)=(32)+(4+3)=+7.ijijiijjijij

Further Discussion

We can also consider adding these vectors graphically. From our starting point we move +3 in the 𝑥-direction and +4 in the 𝑦-direction. This represents the vector 𝐴𝐵. From this point, we move 2 in the 𝑥-direction and +3 in the 𝑦-direction. This represents the vector 𝑋𝑌. This is shown in the figure on the left.

Then, the sum of 𝐴𝐵 and 𝑋𝑌 is the vector which describes the movement from the original start point to the end point. As the figure on the right shows, the vector 𝐴𝐵+𝑋𝑌 is equal to ij+7.

Example 6: Subtracting Vectors Given as Sums of Unit Vectors

Let 𝐴𝐵=34ij and 𝑋𝑌=4+7ij. Find 𝐴𝐵𝑋𝑌.

Answer

In the same way as when adding, to subtract vectors we consider the i and j components separately.

For the i components, we have 3(4)ii and for the j components we have 47jj. Writing this in full, we get 𝐴𝐵𝑋𝑌=(34)(4+7)=3(4)47=(3(4))+(47)=(3+4)+(11)=711.ijijiijjijijij

Key Points

  1. There are two ways to write vectors when considering horizontal and vertical components: in addition to writing vectors in the form 𝑎,𝑏, we can describe them using sums of the unit vectors i and j. The 𝑥-component 𝑎 is the coefficient of i and the 𝑦-component 𝑏 is the coefficient of j. So, we can write the vector as 𝑎+𝑏ij.
  2. When we add or subtract vectors, we have to combine the i components and the j components separately.
  3. Given two vectors 𝐴𝐵=𝑎+𝑏ij and 𝑋𝑌=𝑐+𝑑ij, to find 𝐴𝐵+𝑋𝑌, add the horizontal components and the vertical components separately.
    So, 𝐴𝐵+𝑋𝑌=𝑝+𝑞ij, where 𝑝=𝑎+𝑐𝑞=𝑏+𝑑.and Written formally, we calculate the sum as follows: (𝑎+𝑏)+(𝑐+𝑑)=(𝑎+𝑐)+(𝑏+𝑑)=(𝑎+𝑐)+(𝑏+𝑑).ijijiijjij Similarly, we can find the difference of the two vectors 𝐴𝐵𝑋𝑌 by taking the difference of their horizontal and vertical components. Hence, 𝐴𝐵𝑋𝑌=𝑝+𝑞ij , where 𝑝=𝑎𝑐𝑞=𝑏𝑑.and Equivalently, this can be written as (𝑎+𝑏)(𝑐+𝑑)=(𝑎𝑐)+(𝑏𝑑)=(𝑎𝑐)+(𝑏𝑑).ijijiijjij

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