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 and are represented as arrows between point and point and between point and point 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 with “4 units right, 2 units up” as we have both the direction and the distance: if we start at , these instructions lead us directly to . Similarly, vector 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 , its first (horizontal) component is ,
- goes , its second (vertical) component is .
Definition: Components of a Vector
The components of a vector are written as , where describes the movement and the 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 for and . Write in the form .
To find the horizontal and vertical components of , that is, to find and such that , we can consider the horizontal and vertical distances from to . Another way to look at this is to find how far we have to move in the horizontal direction () and the vertical direction () to get from to .
The horizontal distance is equal to the difference between the -coordinates; this is . The vertical distance is equal to the difference between the -coordinates, which is .
Hence, we can write the vector from to as
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 , for and , in the form , first calculate the difference between the -coordinates to obtain and then calculate the difference between the -coordinates to obtain :
Example 2: Finding the Horizontal and Vertical Components of a Vector
Consider the vector in the diagram.
- What are the coordinates of its terminal point?
- What are the coordinates of its initial point?
- What are the components of the vector?
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 is the terminal point.
Similarly, the initial point is the point from which the arrow starts. Hence, by looking at the diagram, we can see that is the initial point.
The first component of the vector can be found by considering the difference between the -coordinates of the terminal and initial points; that is, the first component (or, equivalently, the -component) of the vector is . As for the second component, it is the difference in the -coordinates of the terminal and initial points. Hence, the second component (or, equivalently, the -component) is given by . When we write vectors in component form, we use the notation .
Next, we define two special vectors which have length 1. The vector represents moving a distance of positive one in the -direction, and the vector represents moving a distance of positive one in the -direction.
Definition: The Unit Vectors
Using the notation we described earlier, we can define where represents moving one unit in the -direction and no units in the -direction and represents moving no units in the -direction and one unit in the -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 can represent moving from to since the -coordinate increases by one and the -coordinate does not change.
The unit vector can represent moving from to since the -coordinate does not change and the -coordinate increases by one.
The vector represents moving along copies of . We can write which means that we move units in the horizontal direction and no units in the vertical direction.
The vector represents moving along copies of . We can write which means that we move units in the vertical direction and no units in the horizontal direction.
For example, shown in the figure below is the vector .
Once we have vectors of the form and we can add them together to describe any vector in the form .
How To: Writing a Vector as the Sum of Unit Vectors
In general, if we travel from a starting point to a finishing point , then this describes a vector that represents traveling a distance of in the -direction and then a distance of in the -direction.
We can write this vector in two ways: or
Example 3: Writing Vectors as the Sum of Unit Vectors
Using that each square in the grid has length 1, write the vector in the form and then in the form .
From the start point , we move units in the horizontal direction (which represents the vector ) and then move units in the vertical direction (which represents the vector ) to get to the point .
The vector , which represents traveling directly from to , is then the sum of these unit vectors.
Using the vectors and 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 and represent moving to the left and down respectively.
For example, the vector , which represents traveling two negative units in the -direction and three negative units in the -direction, as seen in the figure above, can be written as or
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 and are two unit vectors in the east and north directions respectively. Express its displacement in terms of the two unit vectors and .
We have been told that represents the easterly direction and the northerly direction. The body moved east. Therefore, the vector expressing this displacement will have no northerly component. Therefore, the coefficient of will be zero. We know that it moved 190 cm in the easterly direction; therefore, the coefficient of will be 190. Therefore, the displacement of the body can be written as
Remember that equivalent vectors are vectors that have the same direction and the same magnitude.
Consider now vectors , , , and in the following diagram.
We can see that all vectors lie either on the same line or on parallel lines. However, 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 -component to their -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 - and -axes (such a right triangle is shown for ) and applying the Pythagorean theorem. The lengths of the legs are given by the absolute values of the components of the vectors. We find
We notice that the components of are the opposite of the components of and , which means that they have same magnitude but opposite directions.
The two vectors with the same direction and magnitude are and ; 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 and . As and have the same direction, the ratio of their components must be the same, and their components must have the same sign. Hence, we have
In addition, and have the same magnitude. This means that (thinking of the right triangles we have used above)
Substituting the above expressions for and into this equation, we find that
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 , , and have coordinates , , and respectively. Given that and are equivalent vectors, find the coordinates of .
We are told that and are equivalent vectors, which means that they have the same components. Let us therefore find the components of :
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 to .
As and have same components, we have
Since , it gives us
Substituting in the values for and , we have and
The coordinates of point are .
Let us summarize what we have learned in this explainer.
- The components of a vector are written as , where describes the movement and the movement from the initial point to the terminal point of the vector.
- The components of vector , where and , are given by
- The unit vectors are defined as
- Any vector of components can be written in terms of the unit vectors and as follows:
- Vectors with the same components are equivalent, and, inversely, equivalent vectors have the same components.