Lesson Video: Scalars, Vectors, and Directed Line Segments | Nagwa Lesson Video: Scalars, Vectors, and Directed Line Segments | Nagwa

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

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

16:05

Video Transcript

In this video, we will learn how to recognize, construct, and express directed line segments. We will begin by introducing and recapping some key terms.

A scalar is a quantity that is fully described by a magnitude. For example, length, time, distance, and speed are all scalar quantities. 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 will 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 with a half arrow above it and can be represented graphically as shown. Note that line segment 𝐀𝐁 is different to line segment 𝐁𝐀, which would mean that 𝐵 is the initial point and 𝐴 is the terminal point. We note that the magnitude, also called the norm, of the directed line segment 𝐀𝐁 is just the length of the line segment 𝐀𝐁, which is denoted in either of the two ways shown.

Note that since 𝐁𝐀 lies along the same line segment as 𝐀𝐁, we can conclude that it has the same magnitude. Additionally, directed line segments can be said to be equivalent, which is defined as follows. If two directed line segments have the same magnitude and direction, they are equivalent. As an example of this, consider the parallelogram 𝐴𝐵𝐶𝐷. Since the directed line segment 𝐀𝐁 has the same magnitude and direction as 𝐃𝐂, they are equivalent. The same applies to line segments 𝐀𝐃 and 𝐁𝐂. Let’s now consider an example where we need to apply the idea of equivalent directed line segments.

In the diagram, which of the following directed line segments is equivalent to 𝐀𝐁? Is it (A) 𝐅𝐂, (B) 𝐃𝐄, (C) 𝐁𝐂, (D) 𝐆𝐂, or (E) 𝐆𝐅?

In this question, we’ve been given several directed line segments. In each case, they can be identified by their initial point and their terminal point. For instance, line segment 𝐀𝐁 starts at point 𝐴 and goes to point 𝐵 as highlighted in the diagram. 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 that we need to identify which of the options has the same length as 𝐀𝐁 and goes in the same direction, that is, horizontally from left to right. Let us consider them one by one.

For option (A), 𝐅𝐂 goes in the correct direction, but its length is double that of 𝐀𝐁, so it cannot be equivalent. For options (B) and (E), 𝐃𝐄 and 𝐆𝐅 have the same magnitude as 𝐀𝐁, and the line segments are horizontal. But they are both in the opposite direction, so they can be excluded. In option (C), 𝐁𝐂 has the same magnitude but the direction is completely different, so it cannot be equivalent. However, for option (D), we see that 𝐆𝐂 does indeed have the same direction and magnitude as 𝐀𝐁. Hence, the correct answer is option (D). The directed line segment 𝐆𝐂 is equivalent to 𝐀𝐁.

We will now consider vectors. 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. The 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 three vectors shown. 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 equal to four, negative two, we could present this graphically as a directed line segment. Another vector 𝐛 is given as eight, negative four. 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 or equal to two 𝐚. 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 when 𝑘 is equal to negative one. Negative 𝐚 is equal to negative one multiplied by four, negative two, which is equal to negative one multiplied by four, negative one multiplied by negative two and therefore negative four, two. This can be seen graphically as shown. The two vectors 𝐚 and negative 𝐚 are parallel and have equal magnitude but have opposite directions.

Much like with directed line segments, we can define the idea of equivalent vectors. Two vectors are equivalent if they have the same magnitude and direction or if all of their corresponding components are equal and of the same dimension. We can also define opposite vectors. Two vectors are opposite if they have the same magnitude but opposite direction.

We will now 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. Vector 𝐯 has a horizontal change of six units and a vertical change of negative three units. Vector 𝐯 can therefore be written six, negative three as shown. We can use the coordinates of the endpoints of a vector to find the horizontal and vertical components of a vector. For any coordinates 𝐴: 𝑥 sub 𝐴, 𝑦 sub 𝐴 and 𝐵: 𝑥 sub 𝐵, 𝑦 sub 𝐵, vector 𝐀𝐁 is equal to 𝑥 sub 𝐵 minus 𝑥 sub 𝐴, 𝑦 sub 𝐵 minus 𝑦 sub 𝐴.

Note that we use the given notation to represent the vector between 𝐴 and 𝐵 even though this is technically a directed line segment. Really what this means is that we are referring to the vector that can be defined by the directed line segment 𝐀𝐁. We often say that a vector has an initial and terminal point. But as with the example shown here, we are simply defining a vector using these points. In reality, one vector can be used to represent a group of all of the equivalent directed line segments with the same magnitude and direction. Note that we will continue to use this notation for vectors throughout this video, since it is a very common way of writing them.

If we return to our previous example of the hexagon and let vector 𝐯 be equivalent to the directed line segment 𝐀𝐁, then this same vector can be used to represent the directed line segments 𝐄𝐃, 𝐅𝐆, and 𝐆𝐂, since they all have the same magnitude and direction. We can define any vector 𝐯 without defining its initial and terminal points as we simply need a magnitude and direction to define it. We will now look at an example where we will use some of the properties of vectors that we have considered so far.

Which vector has the same direction as vector 𝐚?

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 𝑥, 𝑦, where 𝑥 represents the horizontal change between the 𝑥-coordinates of its endpoints and 𝑦 represents the vertical change between the 𝑦-coordinates. Vector 𝐚 can be written as 𝐚 is equal to four, two. All vectors in the same direction can be written as 𝑘 multiplied by four, two, with 𝑘 as a positive scalar.

Looking at the other three vectors on the grid, we have 𝐛 is equal to one, negative one; 𝐜 is equal to one, three; and 𝐝 is equal to four, two. The only vector which is in the same direction as 𝐚 is vector 𝐝. In this case, the two vectors 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 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 now consider how to calculate the magnitude of a vector before looking at one final example of how to find the magnitude of a vector represented graphically. To find the magnitude of vector 𝐯, written as shown, that we saw earlier, we use the Pythagorean theorem. This states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The magnitude of a vector 𝑎, 𝑏 is given by the square root of 𝑎 squared plus 𝑏 squared. In our example, the magnitude of vector 𝐯 is equal to the square root of six squared plus negative three squared. This simplifies to the square root of 36 plus nine, which is equal to root 45, or three root five.

Note that for initial and terminal points 𝐴: 𝑥 sub 𝐴, 𝑦 sub 𝐴 and 𝐵: 𝑥 sub 𝐵, 𝑦 sub 𝐵, the magnitude of the vector 𝐀𝐁 is equal to the square root of 𝑥 sub 𝐵 minus 𝑥 sub 𝐴 all squared plus 𝑦 sub 𝐵 minus 𝑦 sub 𝐴 all squared. Let’s now consider that final example.

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

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 the squares in the grid are of unit length. The magnitude of vector 𝐯 is equal to the square root of one squared plus two squared. This simplifies to the square root of one plus four, which equals root five. Thus, the magnitude of the vector 𝐯 is root five.

We will now summarize the key points from this video. 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. To describe a vector, we need either an initial point and a terminal point or its magnitude and direction. A vector 𝐀𝐁 describes the movement from an initial point 𝐴 to the terminal point 𝐵. For any points 𝐴: 𝑥 sub 𝐴, 𝑦 sub 𝐴 and 𝐵: 𝑥 sub 𝐵, 𝑦 sub 𝐵, then vector 𝐀𝐁 is equal to 𝑥 sub 𝐵 minus 𝑥 sub 𝐴, 𝑦 sub 𝐵 minus 𝑦 sub 𝐴. 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 of their corresponding components are equal and of the same dimension. For a nonzero vector 𝐚, the opposite vector negative 𝐚 has the same magnitude as 𝐚 but points in the opposite direction. We can find the magnitude of a vector 𝑎, 𝑏 by finding the square root of 𝑎 squared plus 𝑏 squared. Finally, given the endpoints 𝐴: 𝑥 sub 𝐴, 𝑦 sub 𝐴 and 𝐵: 𝑥 sub 𝐵, 𝑦 sub 𝐵 of any vector 𝐀𝐁, we can calculate its magnitude as the square root of 𝑥 sub 𝐵 minus 𝑥 sub 𝐴 all squared plus 𝑦 sub 𝐵 minus 𝑦 sub 𝐴 all squared.

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