Lesson Explainer: Scalars, Vectors, and Directed Line Segments Mathematics

In this explainer, we will learn how to recognize, construct, and express directed line segments.

We can begin by introducing and recapping some key terms.

Definition: Scalar

A scalar is a quantity that is fully described by a magnitude.

For example, length, time, distance, and speed are all scalar quantities.

Recap: Line Segment

A line segment is a part of a line that is bounded by two distinct endpoints and contains every point on the line between its endpoints.

We now consider a directed line segment, when one of the endpoints is an initial point, and the other one is a terminal point. If 𝐴 is the initial point and 𝐵 is the terminal point, then the directed line segment is written 𝐴𝐵 and can be represented graphically as follows.

Note that 𝐴𝐵 is different than 𝐵𝐴, which would mean that 𝐵 is the initial point and 𝐴 is the terminal point.

We will now consider vectors.

Definition: Vector

A vector is an object that has a magnitude and a direction.

Displacement, velocity, and acceleration are all examples of vector quantities.

Vectors can be represented graphically using a directed line segment. This direction of the line segment represents the direction of the vector, and the length of the line segment represents the magnitude of the vector.

Consider the following three vectors.

As these three vectors have the same magnitude and direction, we can say that they are equivalent, or equal. Equal vectors may have different endpoints.

We will now consider how we can multiply a vector by a scalar quantity. If we have a vector 𝑎=(4,2), we could present this graphically as a directed line segment.

Another vector, 𝑏, is given as (8,4).

Vectors 𝑎 and 𝑏 are parallel and have the same direction. However, vector 𝑏 is twice the magnitude of vector 𝑎. We could say that 𝑏 is equivalent to 2𝑎, 𝑏=2𝑎. Note that each of the 𝑥-and 𝑦-components of vector 𝑎 are doubled to give those of vector 𝑏.

We can multiply any vector, 𝑣, by any scalar quantity, 𝑘, to create a vector, 𝑘𝑣, which is parallel to vector 𝑣.

Consider what happens if 𝑘=1. Then, 𝑎=1(4,2)=(1×4,1×(2))=(4,2).

We can show this in the following diagram.

The two vectors 𝑎 and 𝑎 are parallel and have equal magnitude but have opposite directions.

Definition: Equivalent Vectors

Two vectors are equivalent if they have the same magnitude and direction or if all their corresponding components are equal and are of the same dimension.

We can also define opposite vectors.

Definition: Opposite Vectors

Two vectors are opposite if they have the same magnitude but opposite direction.

We can consider a vector 𝑣, which has magnitude and direction as shown by the length of the line segment and the arrow.

We can represent this vector in terms of the horizontal and vertical change. In the form (𝑎,𝑏), 𝑎 represents the horizontal change between the 𝑥-coordinates of its endpoints, and 𝑏 represents the vertical change between the 𝑦-coordinates of its endpoints. Alternatively, a vector can be written in the form 𝑎𝑖+𝑏𝑗, where 𝑖 is the vector in the positive 𝑥-direction of magnitude 1, and 𝑗 is the vector in the positive 𝑦-direction of magnitude 1.

Vector 𝑣 has a horizontal change of 6 units and a vertical change of 3 units and, therefore, can be written as (6,3). Note that if the movement is to the left, the horizontal change is negative, and, similarly, if the movement is downward, the vertical change is negative.

We can use the coordinates of the endpoints of a vector to find the horizontal and vertical components of a vector.

Definition: The Horizontal and Vertical Components of a Vector Using Endpoints

For any coordinates 𝐴=(𝑥,𝑦) and 𝐵=(𝑥,𝑦), 𝐴𝐵=(𝑥𝑥,𝑦𝑦).

To find the magnitude of a vector, 𝑣, written with the notation 𝑣, we use the Pythagorean theorem. This theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Definition: Magnitude of a Vector

The magnitude of a vector (𝑎,𝑏) is given by (𝑎,𝑏)=𝑎+𝑏.

The magnitude of vector 𝑣=(6,3) above can be found by 𝑣=6+(3)=36+9=45.

We can now look at some questions where we consider many different aspects of vectors, including their representation and their attributes. We will start by looking at the information that we would need to describe a vector.

Example 1: Identifying the Information Needed to Define a Vector

What information do you need to fully define a vector?

Answer

A vector is an object that has a magnitude and a direction. Therefore, we could describe a vector by stating its magnitude and direction. However, we can consider whether there are other ways we could convey this information indirectly.

Let us consider the following representation of a vector.

We could represent the magnitude and direction of a vector graphically with two endpoints. For example, if a vector was given to us with endpoints at the coordinates 𝐴(3,4) and 𝐵(2,1), we could represent this as follows.

However, the problem with having just the endpoints of the graphical representation of the vector is that we do not know which direction the travel undertakes: is it from the coordinate 𝐴 to 𝐵 or from the coordinate 𝐵 to 𝐴? But if we knew which point was the initial point and which was the terminal point, we could establish the exact vector. Thus, having this information allows us to represent a vector.

In this example, if 𝐴(3,4) is the initial point and 𝐵(2,1) is the terminal point, then the vector 𝐴𝐵 would be as shown.

If we are given the initial and terminal points of a vector as 𝐴(𝑥,𝑦) and 𝐵(𝑥,𝑦), we can calculate the magnitude of the vector 𝐴𝐵, written as 𝐴𝐵, using the Pythagorean theorem: 𝐴𝐵=(𝑥𝑥)+(𝑦𝑦).

Given an initial point and a terminal point of a vector, the direction of the vector is from the initial point to the terminal point.

Therefore, we can fully define a vector with either the initial point and terminal point or the magnitude and direction.

We can note the formula used in the previous example.

Definition: Magnitude of a Vector Given Its Endpoints

For initial and terminal points of a vector, 𝐴(𝑥,𝑦) and 𝐵(𝑥,𝑦), the magnitude of the vector 𝐴𝐵 is given by 𝐴𝐵=(𝑥𝑥)+(𝑦𝑦).

In the next example, we will see how we identify vectors with the same direction.

Example 2: Identifying Vectors with the Same Direction

Which vector has the same direction as 𝑎?

Answer

We can begin by noting that two vectors are in the same direction if one is a positive scalar multiple of the other.

We can write all of the vectors in the form (𝑎,𝑏), where 𝑎 represents the horizontal change between the 𝑥-coordinates of its endpoints and 𝑏 represents the vertical change between the 𝑦-coordinates.

Vector 𝑎 can be written as 𝑎=(4,2).

All vectors in the same direction can be written as 𝑘(4,2), with 𝑘 as a positive scalar. For example, the vector (8,4), with 𝑘=2, would be in the same direction, as would the vector (2,1), 𝑘=12.

Looking at the other vectors given on the grid, we can write that 𝑏=(1,1),𝑐=(1,3),𝑑=(4,2).

The only vector which is in the same direction as 𝑎=(4,2) is 𝑑=(4,2). In this case, 𝑎 and 𝑑 are the same, even though they have different initial and terminal points. This means they have the same magnitude and direction.

Although not required for this question, we can recognize that vectors 𝑎 and 𝑑 are also equal vectors, as they have the same magnitude and direction.

Thus, we have identified that the vector with the same direction as 𝑎 is vector 𝑑.

We will see in the following example how the choice of terminal point and initial point in vector notation is both important and useful when modeling vectors.

Example 3: Identifying the Terminal Point of a Vector

What is the terminal point of the vector 𝐴𝐵?

Answer

A vector 𝐴𝐵 can be represented in the following way.

The ordering of the points and direction of the arrow, in the form 𝐴𝐵, are indicative of the movement of the vector. In this case, we travel from 𝐴 to 𝐵, with 𝐴 being the initial point and 𝐵 being the terminal point.

Therefore, the terminal point of vector 𝐴𝐵 is 𝐵.

We will now look at an example of how to find the magnitude of a vector represented graphically.

Example 4: Finding the Magnitude of a Vector

Find the magnitude of the vector 𝑣 shown on the grid of unit squares below.

Answer

The magnitude of a vector represented graphically is the length of the line segment. We can calculate the magnitude of the vector 𝑣 by using the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

We consider the horizontal and vertical changes between the initial point and terminal point, given that the squares in the grid are of unit length.

We can write that 𝑣=1+2=5.

Thus, the magnitude of the vector 𝑣 is 5.

We will now look an example involving equivalent vectors.

Example 5: Identifying the Properties of Equivalent Vectors

Select all the statements that must be true if 𝑢 and 𝑣 are equivalent vectors.

  1. 𝑢 and 𝑣 have the same initial point.
  2. 𝑢 and 𝑣 have the same terminal point.
  3. 𝑢=𝑣
  4. The initial point of 𝑣 is the terminal point of 𝑢.
  5. The initial point of 𝑢 is the terminal point of 𝑣.

Answer

We recall that two vectors are equivalent if they have the same magnitude and direction.

The statement given in option C, 𝑢=𝑣, is that the magnitudes of 𝑢 and 𝑣 are equal; therefore, this is a true statement.

We can check if any of the other statements would also apply. To do this, we can consider a graphical representation of some vectors 𝑢 and 𝑣. We can take this vector 𝑢 which has a magnitude of 1 in the direction of the positive 𝑥-axis. A vector 𝑣 of equal magnitude could be drawn parallel to 𝑢 and pointing in the same direction.

Vectors 𝑢 and 𝑣 are equivalent because they have the same magnitude and direction. However, we can see that these vectors do not have the same initial or terminal point, nor is the terminal point of one the initial point of the other. Therefore, while the statements given in options A, B, D, and E may be true in some situations, they are not true of all equivalent vectors.

We can give the answer that the statement which is true for equivalent vectors is 𝑢=𝑣.

We will now see how we can use our understanding of the horizontal and vertical changes of a vector, along with information about one of the endpoints, to find the other endpoint.

Example 6: Finding the Initial Point given a Terminal Point and a Vector

If 𝐴𝐵=𝑖2𝑗 and 𝐵=(4,3), then the coordinates of 𝐴 are .

  1. (5,1)
  2. (3,5)
  3. (3,5)
  4. (5,1)

Answer

In this question, we are given the information about a vector 𝐴𝐵, a vector which has an initial point at 𝐴 and a terminal point at 𝐵. We are also given the coordinates of 𝐵.

We can begin by considering a graphical representation of 𝐴𝐵=𝑖2𝑗. This vector has a horizontal change of 1 and a vertical change of 2. We could also write 𝐴𝐵 as (1,2).

As the terminal point 𝐵 is at (4,3), then we can represent this point and the vector as follows.

Using this, we can see that the coordinates of 𝐴 are (3,5).

Alternatively, without graphing, we recognize that the horizontal and vertical components of a vector are given by subtracting the coordinates of the initial point from those of the terminal point.

For any coordinates 𝐴=(𝑥,𝑦) and 𝐵=(𝑥,𝑦), 𝐴𝐵=(𝑥𝑥,𝑦𝑦).

We can substitute 𝐴𝐵=(1,2) and 𝐵=(4,3) into this equation, giving (1,2)=(4𝑥,3𝑦).

Two vectors are equal if their horizontal and vertical components are equal. Therefore, we equate the horizontal components to give 1=4𝑥1+𝑥=4𝑥=3.

Similarly, we evaluate the 𝑦-components, giving 2=3𝑦2+𝑦=3𝑦=5.

Therefore, 𝐴=(𝑥,𝑦) can be given as (3,5).

In the following example, we will look at opposite vectors.

Example 7: Identifying the Properties of Opposite Vectors

If 𝐴 is a nonzero vector, then .

  1. 𝐴 and 𝐴 have the same direction
  2. 𝐴 and 𝐴 have opposite directions
  3. 𝐴𝐴
  4. 𝐴>𝐴

Answer

To answer this question, we consider the nonzero vectors 𝐴 and 𝐴. The vector 𝐴 has the same magnitude as 𝐴 but points in the opposite direction.

Therefore, we can complete the question statement: ifisanonzerovector,thenandhaveoppositedirections𝐴𝐴𝐴.

We can consider that answer option C, 𝐴𝐴 (𝐴 is perpendicular to 𝐴), cannot be true, as vectors 𝐴 and 𝐴 are parallel. Similarly, the statement in option D, 𝐴>𝐴, is false. This is because opposite vectors have the same magnitude, so we could write 𝐴=𝐴.

In the final example, we will apply our knowledge of vectors to help us solve a geometry problem.

Example 8: Finding the Shape Formed by Four Given Vectors

What shape is formed by these vectors?

Answer

In the diagram, we note that there are two common vectors, 𝑢, and 𝑣. Both 𝐴𝐵 and 𝐶𝐷 are written with the vector 𝑢. Similarly, 𝐵𝐷 and 𝐴𝐶 are both labeled with vector 𝑣. When two vectors are equal, they will have the same magnitude and direction.

This demonstrates that in this diagram the opposite sides 𝐴𝐵 and 𝐶𝐷 will be the same length and parallel.

The other pair of opposite sides, 𝐵𝐷 and 𝐴𝐶, will also be the same length and parallel.

A quadrilateral with parallel opposite sides is defined to be a parallelogram.

Therefore, we can give the solution that the shape formed by these vectors is a parallelogram.

We will now summarize the key points.

Key Points

  • A vector is an object that has a magnitude and a direction. We can represent it as a directed line segment, with the length representing the magnitude and the arrow representing the direction.
  • To describe a vector, we need either an initial point and terminal point, or its magnitude and direction.
  • A vector 𝐴𝐵 describes the movement from the initial point, 𝐴, to the terminal point, 𝐵.
  • For any points 𝐴=(𝑥,𝑦) and 𝐵=(𝑥,𝑦), 𝐴𝐵=(𝑥𝑥,𝑦𝑦).
  • Two vectors have the same direction if one is a positive scalar multiple of the other.
  • Two vectors are equivalent if they have the same magnitude and direction or if all their corresponding components are equal and are of the same dimension.
  • For a nonzero vector 𝐴, the opposite vector, 𝐴, has the same magnitude as 𝐴 but points in the opposite direction.
  • We can find the magnitude of a vector (𝑎,𝑏) by (𝑎,𝑏)=𝑎+𝑏.
  • Given the endpoints 𝐴(𝑥,𝑦) and 𝐵(𝑥,𝑦) of any vector 𝐴𝐵, we can calculate its magnitude by 𝐴𝐵=(𝑥𝑥)+(𝑦𝑦).

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