Video Transcript
In this video, we will learn how to recognize, construct, and express directed line segments. We will begin by describing a vector connecting two points. Letโs consider the following question.
What information do you need to fully define a vector? Letโs begin by considering the two-dimensional ๐ฅ๐ฆ plane as shown. The arrow drawn represents a vector. There are two ways that we could define this. Firstly, any vector has both magnitude and direction. The length of the line segment drawn is the magnitude of the vector, and the arrow indicates the direction. The start point for a vector is sometimes known as its tail, and the endpoint is known as the head. The direction of any vector is, therefore, from its tail to its head. This means that we also have a second way of defining a vector if we know its initial point or tail and terminal point or head. We can, therefore, conclude that there are two pieces of information that we need to define a vector, either its magnitude and direction or its initial and terminal points.
We will now briefly consider the notation we use when dealing with vectors. If we let the vector drawn be vector ๐ฏ, this is written with a half arrow above the letter. To get from the tail of the vector to its head, we move seven units in the ๐ฅ-direction and three units in the ๐ฆ-direction. This can be written in triangular brackets as shown. Alternatively, we can write a vector in terms of unit vectors ๐ข and ๐ฃ, in this case seven ๐ข plus three ๐ฃ. The component of ๐ข is the movement in the ๐ฅ-direction, and the component of ๐ฃ is the movement in the ๐ฆ-direction. We could also write this as a column vector seven, three as shown.
We denote the magnitude of vector ๐ฏ using absolute value bars, and we can calculate it using the Pythagorean theorem. In this case, the magnitude of vector ๐ฏ is equal to the square root of seven squared plus three squared. We find the sum of the squares of the ๐ฅ- and ๐ฆ-components and then square root the answer. In this example, the magnitude of vector ๐ฏ is equal to the square root of 58.
As already mentioned, we can also define a vector using its initial point and terminal point. In this example, point ๐ด has coordinates negative three, one and point ๐ต has coordinates four, four. We can calculate the vector of the line segment ๐ด๐ต by subtracting the ๐ฅ-coordinates and then subtracting the ๐ฆ-coordinates. Subtracting the ๐ฅ-coordinate of ๐ด from the ๐ฅ-coordinate of ๐ต gives us four minus negative three. With the ๐ฆ-coordinates, we get four minus one. This once again proves that the vector in our diagram is seven, three.
We will now look at a question where we need to identify vectors with the same direction.
Which vector has the same direction as vector ๐?
If two vectors have the same direction on a coordinate plane, then the lines must be parallel. This means that they never meet. So, it is quite obvious from the diagram that the vector that has the same direction as ๐ is vector ๐. Vectors ๐ and ๐ have the same initial point. Therefore, they cannot be in the same direction. In a similar way, vectors ๐ and ๐ have the same terminal point. This means that they cannot be in the same direction. If two line segments have the same initial point or terminal point, they cannot be parallel. Therefore, the vectors cannot be in the same direction.
We can actually go one stage further in this question. We can see from the grid that vector ๐ is equal to four, two. From the initial point to the terminal point, we move four units right and two units up. This is also true of vector ๐. We can, therefore, conclude that vectors ๐ and ๐ have the same magnitude as well as the same direction. When two vectors have the same ๐ฅ- and ๐ฆ-component, they will have the same magnitude and direction.
In our next question, we need to identify the endpoint of a vector.
What is the terminal point of the vector ๐๐?
Vector ๐๐ is a line segment that starts at point ๐ด and ends at point ๐ต. Point ๐ด is known as the initial point or tail of the vector. Point ๐ต is known as the terminal or endpoint of the vector, often referred to as the head. Writing ๐๐ with a half arrow above it is common notation for the vector that starts at point ๐ด and finishes or terminates at point ๐ต. We can, therefore, conclude that point ๐ต is the terminal point of the vector.
In our next question, we will calculate the magnitude of a vector.
Find the magnitude of the vector ๐ฏ shown on the grid of unit squares below.
Any vector, in this case ๐ฏ, can be written in terms of its ๐ฅ- and ๐ฆ-components. Any vector will have an initial or start point and a terminal or endpoint. To get from the initial point to the terminal point, we move one unit right and two units up. This means that vector ๐ฏ is equal to one, two. The magnitude of any vector is denoted by absolute value bars. The magnitude is equal to the length of the line segment and can be calculated by finding the sum of the squares of the ๐ฅ- and ๐ฆ-components and then square rooting the answer. In this question, the magnitude of vector ๐ฏ is equal to the square root of one squared plus two squared. One squared is equal to one, and two squared is equal to four. This means that the magnitude of vector ๐ฏ is equal to the square root of five.
In our final question, we will identify the shape formed by four vectors.
What shape is formed by these vectors?
We notice immediately from the diagram that we have two ๐ฎ vectors, from point ๐ด to point ๐ต and also from point ๐ถ to point ๐ท. These vectors have the same magnitude and direction. This means that theyโre parallel and equal in length. In the same way, we see that the line segments ๐ด๐ถ and ๐ต๐ท are equal to vector ๐ฏ. These two sides of the shape must therefore also be parallel and equal in length. We know that any four-sided shape or quadrilateral that is formed by two sets of equal-length parallel sides is called a parallelogram. This means that, based on the information given on the diagram, this shape is a parallelogram. There are special types of parallelograms, such as rectangles, squares, and rhombuses. However, we do not have enough information in this question to prove that our parallelogram is a rectangle, square, or rhombus.
We will now summarize the key points from this video. We established at the start of this video that a vector must have a magnitude and a direction. This can be pictured as a directed line segment, as shown. The length of the line is the magnitude of the vector, and the arrow indicates the direction. Vectors in two dimensions have an ๐ฅ- and ๐ฆ-component. As this vector is moving right and down, it will have a positive ๐ฅ-component and a negative ๐ฆ-component. The vector moves five units to the right and four units down. Therefore, its components are five, negative four. We also know that every vector has an initial point and a terminal point. The initial point is also sometimes called the tail of the vector, and the terminal point is the head of the vector.
We also saw that if vector ๐ has component ๐ฅ one, ๐ฆ one and vector ๐ has components ๐ฅ two, ๐ฆ two, then vector ๐๐ is equal to vector ๐ minus vector ๐. To calculate vector ๐๐, we can subtract the ๐ฅ- and ๐ฆ-components separately. Finally, we saw that the magnitude of vector ๐ฏ with components ๐ฅ, ๐ฆ is equal to the square root of ๐ฅ squared plus ๐ฆ squared. We denote the magnitude with absolute value bars, and it is equal to the sum of the squares of the two components square rooted.
