In this explainer, we will learn how to write vectors in component form using fundamental unit vectors.
We know that there are two components to a vector in two dimensions, which are called the - and -components, with being horizontal and being vertical. Given the components of a 2D vector, we can express it in its component form. For instance, if the - and -components of a 2D vector are equal to and , respectively, the component form of the vector is given by . We can represent this vector on the Cartesian plane by an arrow that starts from the origin and ends at the point , as shown below.
Another method of expressing a 2D vector uses special vectors known as fundamental unit vectors.
Definition: Fundamental Unit Vectors
In two dimensions, the fundamental unit vectors, denoted and , are horizontal and vertical unit vectors, respectively, with nonnegative components. The fundamental unit vectors in component form are given by
We note that the fundamental unit vectors and only have one nonzero component and that the nonzero component for both vectors is equal to 1. From the component form of given above, we can represent it by an arrow beginning at the origin and ending at the point , which lies on the positive -axis.
Similarly, from the component form of , we can represent it on the Cartesian coordinate as follows.
In other words, and lie parallel to the - and -axes, respectively, pointing toward the positive direction of the respective axes.
In our first example, we will consider how to express a vertical vector in terms of the fundamental unit vectors.
Example 1: Expressing a 2D Vector in terms of Fundamental Unit Vectors
Given that , express the vector in terms of the unit vectors and .
Answer
In this example, we are given a vector in component form, which we need to express in terms of the (fundamental) unit vectors and . Recall that we can represent a vector by an arrow that starts from the origin and ends at the point on the Cartesian plane. Since the component form of is , we represent it by an arrow from the origin to the point .
Let us consider how to express this in terms of and . We recall the component forms of and :
Hence, they are represented on the Cartesian plane as follows.
In particular, we can see that is a horizontal vector, while is a vertical vector. Since our vector is purely vertical, we only need to use to express this vector. Let us first observe how to achieve this through the following graph.
By stacking two copies of on top of each other, we can produce . This tells us that
In the previous example, we expressed a vertical 2D vector in terms of the fundamental unit vector . Similar reasoning leads us to express any horizontal vector in terms of . Now, let us discuss how to express a vector that is neither vertical nor horizontal in terms of the fundamental unit vectors.
Example 2: Expressing a 2D Vector in terms of Fundamental Unit Vectors
The given figure shows a vector in a plane. Express this vector in terms of the unit vectors and .
Answer
In this example, we need to express a given vector in terms of the (fundamental) unit vectors and . Let us recall the graphical representations of these vectors on the Cartesian plane.
We can see that is a horizontal vector, while is a vertical vector. Both these vectors have a length of 1 (hence the name unit vectors) and travel toward the positive directions of the - and -axes respectively.
To express in terms of and , we need to consider the - and -components of the vector separately. Let us first consider its -component since it is positive. From the graph, we can see that the -component of is equal to 2. Using the vertical unit vector , we can produce the -component as follows.
As seen above, we can produce the -component of by adding two copies of , which is the same as .
Next, let us consider the -component, which is . Since this component is negative, we need to place a negative sign in front of to reverse the direction of .
This tells us that the -component of is . Adding together both components of and also following the convention to write the -component first, we obtain
In the previous example, we expressed a vector given on a Cartesian plane in terms of fundamental unit vectors and . We can apply this method to any vector with integer-valued components as shown below.
This leads to a general formula that can be used to express a vector in component form in terms of the fundamental unit vectors. While we only justified this formula for vectors with integer-valued components, this formula holds for general vectors. We will assume this for the remainder of this explainer.
Formula: Vectors in terms of Fundamental Unit Vectors
A vector in component form, , can be written in terms of the fundamental unit vectors by
In our next example, we will apply this formula to express a 2D vector in terms of fundamental unit vectors.
Example 3: Expressing the Components of a Vector in terms of the Fundamental Unit Vectors
Express the vector using the unit vectors and .
Answer
In this example, we need to express a vector, given in component form, in terms of the unit vectors and , which are also known as the fundamental unit vectors. Recall that a 2D vector, , can be written in terms of the fundamental unit vectors by
Since our given vector is , we can apply this formula with and to write
In the previous example, we expressed a 2D vector given in component form in terms of fundamental unit vectors. Let us now consider how to achieve this when a vector is given by specifying the beginning and ending points on a coordinate plane.
One way to write this vector in terms of fundamental unit vectors is by first finding the coordinate form of this vector and then converting the form. But we do not have to go through the coordinate form to achieve this. Instead, we can achieve this by identifying the horizontal and vertical components of this vector, so that we can write it as a sum of horizontal and vertical vectors as shown below.
In the next example, we will express a 2D vector given as a graph in a coordinate plane in terms of fundamental unit vectors.
Example 4: Expressing a Vector in terms of the Fundamental Unit Vectors
The given figure shows a vector in a plane. Express this vector in terms of the unit vectors and .
Answer
In this example, we need to write a vector in terms of the unit vectors and when it is given as a graph on a coordinate plane. Recall that these unit vectors are defined by
In other words, these are horizontal and vertical unit vectors, respectively, pointing toward the positive direction of the corresponding axes. Hence, we need to express the given vector as a sum of horizontal and vertical vectors. Let us first graphically identify the horizontal and vertical vectors whose sum equals the given vector.
The horizontal vector shown above spans two grid lengths and points toward the positive direction of the -axis; hence, its horizontal component is equal to . This vector can be written as
The vertical vector in the diagram above spans 10 grid lengths and points toward the positive direction of the -axis, which tells us that its vertical component is equal to . This gives us the vertical vector
We know that adding these two vectors will produce the given vector. Hence, the given vector is equal to
In the previous example, we expressed a 2D vector in terms of the fundamental unit vectors when the vector was given through a graph on a grid between two points. While this approach can always be used, it requires us to graph the points on the coordinate plane. Hence, it is useful to know the formula for achieving this when we are only given the coordinates of the two endpoints of the vector.
Let us consider a vector from point to . In this case, the -component of vector is , and the -component is , as we can see below.
Hence, we can express vector in terms of fundamental unit vectors.
Formula: 2D Vectors between Two Points in terms of Fundamental Unit Vectors
Consider the points and . Then, vector can be written in terms of the fundamental unit vectors by
In our final example, we will apply this formula to write a 2D vector in terms of fundamental unit vectors.
Example 5: Expressing a Vector in terms of the Fundamental Unit Vectors
Given that and , express the vector in terms of the unit vectors and .
Answer
In this example, we need to express a 2D vector in terms of the unit vectors and , which are also known as the fundamental unit vectors, when the vector is given by specifying the coordinates of its endpoints. Recall that vector , with points to , can be written in terms of the fundamental unit vectors by
Since we are given that and , we can apply this formula with
This leads to
Hence,
Let us finish by recapping a few important concepts from this explainer.
Key Points
- The fundamental unit vectors, denoted and , are the
horizontal and vertical unit vectors, respectively, with nonnegative components. The component forms of
and are given by
They can be represented on the Cartesian plane as follows.
- A vector in component form, , can be written in terms of the fundamental unit vectors by
- Consider the points to . Then, vector can be written in terms of the fundamental unit vectors by
