Lesson Explainer: Components of a Vector Mathematics

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

The general definition of a vector is an object defined by its direction and magnitude. Geometrically, it is represented by an arrow (a directed line segment, the arrow indicating the direction) connecting an initial point with a terminal point. The direction of the vector is the direction taken when moving from the initial point to the terminal point, and its magnitude is the distance between the two points (or the length of the segment between the two points).

In the following diagram, vectors 𝐴𝐵 and 𝐶𝐷 are represented as arrows between point 𝐴 and point 𝐵 and between point 𝐶 and point 𝐷 respectively.

Placing our vectors in a coordinate plane allows us to describe the vectors much more easily, namely, with their components. Let us learn what vector components are.

In the coordinate plane, we can fully describe vector 𝐴𝐵 with “4 units right, 2 units up” as we have both the direction and the distance: if we start at 𝐴, these instructions lead us directly to 𝐵. Similarly, vector 𝐶𝐷 can be described with “1 unit left, 1 unit up.”

These descriptions are the basis for what we call the components of a vector. Recall that point coordinates left/right and down/up from the origin are described using negative/positive numbers. We use positive and negative numbers in the same way to describe vectors with their components. If the vector

  • goes to the leftright/, its first (horizontal) component is negativepositive/,
  • goes downup/, its second (vertical) component is negativepositive/.

Definition: Components of a Vector

The components of a vector are written as (𝑎,𝑏), where 𝑎 describes the horizontal movement and 𝑏 the vertical movement from the initial point to the terminal point of the vector.

In the following diagram, various vectors are displayed alongside their components.

Let us see with the next example how to find the components of a vector that is represented in a coordinate plane.

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 6−3=3. The vertical distance is equal to the difference between the 𝑦-coordinates, which is 9−2=7.

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

The way we found the vector components in the previous example using the coordinates of the initial and terminal points of the vector can be summarized as follows.

How To: Writing a Vector in Component Form

To write a vector 𝐴𝐵, for 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=𝑥−𝑥𝑏=𝑦−𝑦.whereand

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 terminal point.

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 −1−2=−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 ⃑𝑖=(1,0)⃑𝑗=(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.

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

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

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

Once we have vectors of the form 𝑎⃑𝑖 and 𝑏⃑𝑗 we can add them together to describe any vector in the form 𝑎⃑𝑖+𝑏⃑𝑗.

How To: Writing 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 that represents traveling a distance of (𝑥−𝑥) in the 𝑥-direction and then a distance of 𝑦−𝑦 in the 𝑦-direction.

We can write this vector in two ways: 𝐶𝐷=(𝑥−𝑥,𝑦−𝑦) or 𝐶𝐷=(𝑥−𝑥)⃑𝑖+(𝑦−𝑦)⃑𝑗.

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 𝑎⃑𝑖+𝑏⃑𝑗 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 2⃑𝑖) and then move +3 units in the vertical direction (which represents the vector 3⃑𝑗) 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=2⃑𝑖+3⃑𝑗=(2,3).

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 to the left and down respectively.

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 (−2⃑𝑖)+(−3⃑𝑗) or 𝐴𝐵=−2⃑𝑖−3⃑𝑗.

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 cm in the easterly direction; therefore, the coefficient of ⃑𝑖 will be 190. Therefore, the displacement of the body can be written as 190⃑𝑖+0⃑𝑗=190⃑𝑖.cm

Remember that equivalent vectors are vectors that have the same direction and the same magnitude.

Consider now vectors 𝐴𝐵=(−2,−1), 𝐶𝐷=(−4,−2), 𝐸𝐹=(−4,−2), and 𝐺𝐻=(4,2) in the following diagram.

We can see that all vectors lie either on the same line or on parallel lines. However, 𝐺𝐻 is not in the same direction as the three others as it points to the right and up, while the others point to the left and down.

It is worth noting here that for vectors to lie on the same line or on parallel lines, the ratios of their 𝑦-component to their 𝑥-component must be the same (it is then the slope of the line).

We can work out the magnitudes of the four vectors by considering the right triangles that one can form using each of them as the hypotenuse with their legs parallel to the 𝑥- and 𝑦-axes (such a right triangle is shown for 𝐴𝐵) and applying the Pythagorean theorem. The lengths of the legs are given by the absolute values of the components of the vectors. We find ‖‖𝐴𝐵‖‖=√2+1=√5,‖‖𝐶𝐷‖‖=√4+2=√20=2√5,‖‖𝐸𝐹‖‖=√4+2=√20=2√5,‖‖𝐺𝐻‖‖=√4+2=√20=2√5.

We notice that the components of 𝐺𝐻 are the opposite of the components of 𝐸𝐹 and 𝐶𝐷, which means that they have same magnitude but opposite directions.

The two vectors with the same direction and magnitude are 𝐶𝐷 and 𝐸𝐹; this is because they have the same components. They are equivalent.

We can easily show that the reverse is true: equivalent vectors (i.e., vectors with same magnitude and direction) must have the same components. Let us consider two equivalent vectors ⃑𝑣=(𝑎,𝑏) and ⃑𝑣=(𝑐,𝑑). As ⃑𝑣 and ⃑𝑣 have the same direction, the ratio of their components must be the same, and their components must have the same sign. Hence, we have 𝑐=𝑘𝑎𝑑=𝑘𝑏𝑘>0.andwith

In addition, ⃑𝑣 and ⃑𝑣 have the same magnitude. This means that (thinking of the right triangles we have used above) √𝑎+𝑏=√𝑐+𝑑.

Substituting the above expressions for 𝑐 and 𝑑 into this equation, we find that √𝑎+𝑏=(𝑘𝑎)+(𝑘𝑏)√𝑎+𝑏=√𝑘𝑎+𝑘𝑏√𝑎+𝑏=√𝑘(𝑎+𝑏)√𝑎+𝑏=𝑘√(𝑎+𝑏).

Hence, 𝑘=1, and (𝑎,𝑏)=(𝑐,𝑑).

Property: Equivalent Vectors and Components

Vectors with the same components are equivalent: they have the same direction and the same magnitude.

Inversely, equivalent vectors have the same components.

Let us use this property in our final example to find the coordinates of a point, given that it is the terminal point of a vector equivalent to another vector.

Example 5: Equivalent Vectors on a Coordinate Plane

The points 𝐴, 𝐵, and 𝐶 have coordinates (−7,1), (−2,4), and (−4,−1) respectively. Given that 𝐴𝐵 and 𝐶𝐷 are equivalent vectors, find the coordinates of 𝐷.

Answer

We are told that 𝐴𝐵 and 𝐶𝐷 are equivalent vectors, which means that they have the same components. Let us therefore find the components of 𝐴𝐵: 𝐴𝐵=(𝑥−𝑥,𝑦−𝑦)𝐴𝐵=(−2−(−7),4−1)𝐴𝐵=(5,3).

We can check that our result is correct using our diagram, that is, that we go 5 units right and 3 units up when going from 𝐴 to 𝐵.

As 𝐴𝐵 and 𝐶𝐷 have same components, we have 𝐶𝐷=(5,3).

Since 𝐶𝐷=(𝑥−𝑥,𝑦−𝑦), it gives us 𝑥−𝑥=5,𝑦−𝑦=3.

Substituting in the values for 𝑥 and 𝑦, we have 𝑥−(−4)=5𝑥=5+(−4)=1, and 𝑦−(−1)=3𝑦=3+(−1)=2.

The coordinates of point 𝐷 are (1,2).

Let us summarize what we have learned in this explainer.

Key Points

  • The components of a vector are written as (𝑎,𝑏), where 𝑎 describes the horizontal movement and 𝑏 the vertical movement from the initial point to the terminal point of the vector.
  • The components (𝑎,𝑏) of vector 𝐴𝐵, where 𝐴(𝑥,𝑦) and 𝐵(𝑥,𝑦), are given by (𝑎,𝑏)=(𝑥−𝑥,𝑦−𝑦).
  • The unit vectors are defined as ⃑𝑖=(1,0)⃑𝑗=(0,1).and
  • Any vector ⃑𝑣 of components (𝑎,𝑏) can be written in terms of the unit vectors ⃑𝑖 and ⃑𝑗 as follows: ⃑𝑣=𝑎⃑𝑖+𝑏⃑𝑗.
  • Vectors with the same components are equivalent, and, inversely, equivalent vectors have the same components.

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