Lesson Video: Scalars, Vectors, and Directed Line Segments

Directed line segment short-cut
Directed line segment short-cut

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.

You are watching: Lesson Video: Scalars, Vectors, and Directed Line Segments. Info created by THVinhTuy selection and synthesis along with other related topics.

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