### 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.