Lesson Explainer: Scalars, Vectors, and Directed Line Segments Mathematics • 12th Grade

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 𝐡 is the terminal point, then the directed line segment is written 𝐴𝐡 and can be represented graphically as follows.

Note that 𝐴𝐡 is different than 𝐡𝐴, which would mean that 𝐡 is the initial point and 𝐴 is the terminal point.

We note that the magnitude (also called the norm) of the directed line segment 𝐴𝐡 is just the length of 𝐴𝐡, which is denoted by ‖‖𝐴𝐡‖‖, or simply 𝐴𝐡. Incidentally, since 𝐡𝐴 lies along the same line segment as 𝐴𝐡, we can conclude that it has the same magnitude. That is, 𝐴𝐡=𝐡𝐴.

Additionally, directed line segments can be said to be equivalent, which is defined as follows.

Definition: Equivalent Directed Line Segments

If two directed line segments have the same magnitude and direction, they are equivalent.

As an example of this, consider a parallelogram 𝐴𝐡𝐢𝐷.

Since the directed line segment 𝐴𝐡 has the same magnitude and direction as 𝐷𝐢, they are equivalent. The same applies to οƒŸπ΅πΆ and 𝐴𝐷.

Let us consider an example where we need to apply the idea of equivalent directed line segments.

Example 1: Identifying Equivalent Directed Line Segments in a Shape

In the diagram, which of the given directed line segments is equivalent to 𝐴𝐡?

  1. οƒŸπΉπΆ
  2. 𝐷𝐸
  3. οƒŸπ΅πΆ
  4. 𝐺𝐢
  5. 𝐺𝐹

Answer

In this question, we have been given several directed line segments. In each case, they can be identified by their initial point and their terminal point. For instance, 𝐴𝐡 starts at point 𝐴 and goes to point 𝐡. We highlight this below.

We recall that a directed line segment is equivalent to another one if it has the same magnitude (i.e., length) and the same direction. This means we need to identify which of the options has the same length as 𝐴𝐡 and goes in the same direction (i.e., horizontally from left to right). Let us go through them one by one.

For option A, οƒŸπΉπΆ goes in the right direction, but its length is double that of 𝐴𝐡, so it cannot be equivalent.

For options B and E, 𝐷𝐸 and 𝐺𝐹 have the same magnitude as 𝐴𝐡, and the line segments are horizontal, but they are both in the opposite direction (i.e., going from right to left), so they can also be excluded.

In option C, οƒŸπ΅πΆ has the same magnitude, but the direction is completely different, so it cannot be equivalent.

However, for option D, we see that 𝐺𝐢 does indeed have the same direction and magnitude as 𝐴𝐡, meaning it must be equivalent. Hence, the correct answer is D.

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. However, unlike directed line segments, vectors do not have a unique starting or ending point. 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 βƒ‘π‘Ž=(4,βˆ’2), we could present this graphically as a directed line segment.

Another vector, ⃑𝑏, is given as (8,βˆ’4).

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

We can multiply any vector, ⃑𝑣, by any scalar quantity, π‘˜, to create a vector, π‘˜βƒ‘π‘£, which is parallel to vector ⃑𝑣.

Consider what happens if π‘˜=βˆ’1. Then, βˆ’βƒ‘π‘Ž=βˆ’1(4,βˆ’2)=(βˆ’1Γ—4,βˆ’1Γ—(βˆ’2))=(βˆ’4,2).

We can show this in the following diagram.

The two vectors βƒ‘π‘Ž and βˆ’βƒ‘π‘Ž are parallel and have equal magnitude but have opposite directions. Much like with directed line segments, we can define the idea of equivalent vectors.

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 (π‘Ž,𝑏), π‘Ž represents the horizontal change between the π‘₯-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 π‘Žβƒ‘π‘–+𝑏⃑𝑗, where ⃑𝑖 is the vector in the positive π‘₯-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 βˆ’3 units and, therefore, can be written as (6,βˆ’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=(π‘₯,𝑦) and 𝐡=(π‘₯,𝑦), 𝐴𝐡=(π‘₯βˆ’π‘₯,π‘¦βˆ’π‘¦).

We note that we can use 𝐴𝐡 to represent the vector between 𝐴 and 𝐡, even though 𝐴𝐡 is technically a directed line segment. We will continue to use this notation for vectors throughout this explainer since it is a very common way of writing vectors.

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 (π‘Ž,𝑏) is given by β€–(π‘Ž,𝑏)β€–=βˆšπ‘Ž+𝑏.

The magnitude of vector ⃑𝑣=(6,βˆ’3) above can be found by ‖‖⃑𝑣‖‖=6+(βˆ’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 2: Identifying the Information Needed to Define a Vector

Five students each present a statement that they think is a sufficient requirement to uniquely define a vector.

Which of their answers are correct?

  1. A magnitude and direction
  2. Two endpoints
  3. An initial point and terminal point
  4. An initial point and magnitude
  5. A direction and terminal point

Answer

To answer this question, let us examine each of the requirements in turn.

In option A, a magnitude and a direction are specified. Recall that a given magnitude by itself is just a number (i.e., a scalar quantity). By adding a direction, we can form a vector as shown.

Thus, option A is correct.

In option B, two endpoints are specified. Let us suppose we were given a vector with endpoints at the coordinates 𝐴(βˆ’3,βˆ’4) and 𝐡(2,1). Then, we could represent it as follows.

However, the problem with just the endpoints of the graphical representation of the vector is that we do not know which direction the travel undertakes: is it from coordinate 𝐴 to coordinate 𝐡 or from coordinate 𝐡 to coordinate 𝐴? This ambiguity means that, by itself, this option is not enough to define a unique vector; so, option B is not correct.

In option C, an initial point and a terminal point are specified. As in option B, we have two endpoints, but this time it is specified that one is initial and one is terminal. This means we know what direction the vector is going and can draw it on the diagram, as shown.

That is to say, 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.

Additionally, if we are given the initial and terminal points of a vector as 𝐴(π‘₯,𝑦) and 𝐡(π‘₯,𝑦), we can calculate the magnitude of the vector 𝐴𝐡, written as ‖‖𝐴𝐡‖‖, using the Pythagorean theorem: ‖‖𝐴𝐡‖‖=(π‘₯βˆ’π‘₯)+(π‘¦βˆ’π‘¦).

Thus, this option allows us to define both a direction and a magnitude, so option C is correct.

For option D, we have to consider an initial point and magnitude. So far, we have found that a magnitude and direction can uniquely define a vector and an initial point and terminal point, but what if we mix the two requirements? Let us explore this possibility with an example. If we consider an arbitrary initial point 𝐴(1,3) and a magnitude of 5, it turns out that we can draw multiple terminal points 𝐡 that satisfy the conditions as follows.

Here, both magnitudes can be calculated in the same way using the Pythagorean theorem: ‖‖𝐴𝐡‖‖=√4+3=5.

Thus, this description of a vector does not result in a unique result. In fact, an infinite number of vectors could be drawn that satisfy this requirement, resulting in a circle centered at (1,3). Therefore, option D is not correct.

For option E, we must consider a direction and terminal point. Once again, we are considering a mix of two valid ways of defining a vector. Let us once again test this with an example. Suppose we had an endpoint of 𝐡(2,1) and a direction pointing from the origin to the endpoint. With this, it would be possible to draw at least 2 vectors, as shown below.

That is to say, if the initial point was at either 𝐴(0,0) or 𝐴(βˆ’2,βˆ’1), the direction and terminal point would be the same. In fact, we could choose any point that lies along this trajectory to be an initial point and it would be valid. Thus, this requirement is not unique and E is not correct.

In conclusion, A and C are the correct options.

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(π‘₯,𝑦) and 𝐡(π‘₯,𝑦), the magnitude of the vector 𝐴𝐡 is given by ‖‖𝐴𝐡‖‖=(π‘₯βˆ’π‘₯)+(π‘¦βˆ’π‘¦).

We note that the calculation is exactly the same for finding the magnitude of a directed line segment.

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

Example 3: Identifying Vectors with the Same Direction

Which vector has the same direction as βƒ‘π‘Ž?

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 (π‘Ž,𝑏), where π‘Ž represents the horizontal change between the π‘₯-coordinates of its endpoints and 𝑏 represents the vertical change between the 𝑦-coordinates.

Vector βƒ‘π‘Ž can be written as βƒ‘π‘Ž=(4,2).

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

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

The only vector which is in the same direction as βƒ‘π‘Ž=(4,2) is ⃑𝑑=(4,2). In this case, βƒ‘π‘Ž 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 βƒ‘π‘Ž 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 βƒ‘π‘Ž 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 4: Identifying the Terminal Point of a Vector

What is the terminal point of the vector 𝐴𝐡?

Answer

A vector 𝐴𝐡 can be represented in the following way.

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

Therefore, the terminal point of vector 𝐴𝐡 is 𝐡.

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

Example 5: 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 6: Identifying the Properties of Equivalent Vectors

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

  1. ⃑𝑒 and ⃑𝑣 have the same initial point.
  2. ⃑𝑒 and ⃑𝑣 have the same terminal point.
  3. ‖‖⃑𝑒‖‖=‖‖⃑𝑣‖‖
  4. The initial point of ⃑𝑣 is the terminal point of ⃑𝑒.
  5. The initial point of ⃑𝑒 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, ‖‖⃑𝑒‖‖=‖‖⃑𝑣‖‖, is that the magnitudes of ⃑𝑒 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 ⃑𝑒 and ⃑𝑣. We can take this vector ⃑𝑒 which has a magnitude of 1 in the direction of the positive π‘₯-axis. A vector ⃑𝑣 of equal magnitude could be drawn parallel to ⃑𝑒 and pointing in the same direction.

Vectors ⃑𝑒 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 ‖‖⃑𝑒‖‖=‖‖⃑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 7: Finding the Initial Point given a Terminal Point and a Vector

Fill in the blank: If 𝐴𝐡=2βƒ‘π‘–βˆ’3⃑𝑗 and 𝐡=(5,6), then the coordinates of 𝐴 are .

  1. (7,3)
  2. (βˆ’3,βˆ’9)
  3. (3,9)
  4. (βˆ’7,βˆ’3)

Answer

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

We can begin by considering a graphical representation of 𝐴𝐡=2βƒ‘π‘–βˆ’3⃑𝑗. This vector has a horizontal change of 2 and a vertical change of βˆ’3. We could also write 𝐴𝐡 as (2,βˆ’3).

As the terminal point 𝐡 is at (5,6), then we can represent this point and the vector as follows.

Using this, we can see that the coordinates of 𝐴 are (3,9).

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=(π‘₯,𝑦) and 𝐡=(π‘₯,𝑦), 𝐴𝐡=(π‘₯βˆ’π‘₯,π‘¦βˆ’π‘¦).

We can substitute 𝐴𝐡=(2,βˆ’3) and 𝐡=(5,6) into this equation, giving (2,βˆ’3)=(5βˆ’π‘₯,6βˆ’π‘¦).

Two vectors are equal if their horizontal and vertical components are equal. Therefore, we equate the horizontal components to give 2=5βˆ’π‘₯2+π‘₯=5π‘₯=3.

Similarly, we evaluate the 𝑦-components, giving βˆ’3=6βˆ’π‘¦βˆ’3+𝑦=6𝑦=9.

Therefore, 𝐴=(π‘₯,𝑦) can be given as (3,9).

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

Example 8: Identifying the Properties of Opposite Vectors

Fill in the blank: If ⃑𝐴 is a nonzero vector, then .

  1. ⃑𝐴 and βˆ’βƒ‘π΄ have the same direction
  2. ⃑𝐴 and βˆ’βƒ‘π΄ have opposite directions
  3. βˆ’βƒ‘π΄βŸ‚βƒ‘π΄
  4. β€–β€–βˆ’βƒ‘π΄β€–β€–<‖‖⃑𝐴‖‖

Answer

To answer this question, we consider the nonzero vectors ⃑𝐴 and βˆ’βƒ‘π΄. The vector βˆ’βƒ‘π΄ has the same magnitude as ⃑𝐴 but points in the opposite direction.

Therefore, we can complete the question statement: ifisanonzerovector,thenandhaveoppositedirectionsβƒ‘π΄βƒ‘π΄βˆ’βƒ‘π΄.

We can consider that answer option C, βˆ’βƒ‘π΄βŸ‚βƒ‘π΄ (βˆ’βƒ‘π΄ is perpendicular to ⃑𝐴), cannot be true, as vectors ⃑𝐴 and βˆ’βƒ‘π΄ are parallel. Similarly, the statement in option D, β€–β€–βˆ’βƒ‘π΄β€–β€–>‖‖⃑𝐴‖‖, is false. This is because opposite vectors have the same magnitude, so we could write β€–β€–βˆ’βƒ‘π΄β€–β€–=‖‖⃑𝐴‖‖.

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

Example 9: 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, ⃑𝑒, and ⃑𝑣. Both 𝐴𝐡 and 𝐢𝐷 are written with the vector ⃑𝑒. Similarly, 𝐡𝐷 and 𝐴𝐢 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 𝐴𝐡 and 𝐢𝐷 will be the same length and parallel.

The other pair of opposite sides, 𝐡𝐷 and 𝐴𝐢, 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 directed line segment is an object with an initial point, a terminal point, and a direction.
  • 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 𝐴𝐡 describes the movement from the initial point, 𝐴, to the terminal point, 𝐡.
  • For any points 𝐴=(π‘₯,𝑦) and 𝐡=(π‘₯,𝑦), 𝐴𝐡=(π‘₯βˆ’π‘₯,π‘¦βˆ’π‘¦).
  • 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, βˆ’βƒ‘π΄, has the same magnitude as ⃑𝐴 but points in the opposite direction.
  • We can find the magnitude of a vector (π‘Ž,𝑏) by β€–(π‘Ž,𝑏)β€–=βˆšπ‘Ž+𝑏.
  • Given the endpoints 𝐴(π‘₯,𝑦) and 𝐡(π‘₯,𝑦) of any vector 𝐴𝐡, we can calculate its magnitude by ‖‖𝐴𝐡‖‖=(π‘₯βˆ’π‘₯)+(π‘¦βˆ’π‘¦).

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