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

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. Let’s consider the following question.

What information do you need to fully define a vector? Let’s begin by considering the two-dimensional 𝑥𝑦 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 𝐯, 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 𝑥-direction and three units in the 𝑦-direction. This can be written in triangular brackets as shown. Alternatively, we can write a vector in terms of unit vectors 𝐢 and 𝐣, in this case seven 𝐢 plus three 𝐣. The component of 𝐢 is the movement in the 𝑥-direction, and the component of 𝐣 is the movement in the 𝑦-direction. We could also write this as a column vector seven, three as shown.

We denote the magnitude of vector 𝐯 using absolute value bars, and we can calculate it using the Pythagorean theorem. In this case, the magnitude of vector 𝐯 is equal to the square root of seven squared plus three squared. We find the sum of the squares of the 𝑥- and 𝑦-components and then square root the answer. In this example, the magnitude of vector 𝐯 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 𝐴 has coordinates negative three, one and point 𝐵 has coordinates four, four. We can calculate the vector of the line segment 𝐴𝐵 by subtracting the 𝑥-coordinates and then subtracting the 𝑦-coordinates. Subtracting the 𝑥-coordinate of 𝐴 from the 𝑥-coordinate of 𝐵 gives us four minus negative three. With the 𝑦-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 𝐚?

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 𝐚 is vector 𝐝. Vectors 𝐚 and 𝐛 have the same initial point. Therefore, they cannot be in the same direction. In a similar way, vectors 𝐚 and 𝐜 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 𝐚 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 𝐝. We can, therefore, conclude that vectors 𝐚 and 𝐝 have the same magnitude as well as the same direction. When two vectors have the same 𝑥- and 𝑦-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 𝐀𝐁?

Vector 𝐀𝐁 is a line segment that starts at point 𝐴 and ends at point 𝐵. Point 𝐴 is known as the initial point or tail of the vector. Point 𝐵 is known as the terminal or endpoint of the vector, often referred to as the head. Writing 𝐀𝐁 with a half arrow above it is common notation for the vector that starts at point 𝐴 and finishes or terminates at point 𝐵. We can, therefore, conclude that point 𝐵 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 𝐯 shown on the grid of unit squares below.

Any vector, in this case 𝐯, can be written in terms of its 𝑥- and 𝑦-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 𝐯 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 𝑥- and 𝑦-components and then square rooting the answer. In this question, the magnitude of vector 𝐯 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 𝐯 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 𝐮 vectors, from point 𝐴 to point 𝐵 and also from point 𝐶 to point 𝐷. These vectors have the same magnitude and direction. This means that they’re parallel and equal in length. In the same way, we see that the line segments 𝐴𝐶 and 𝐵𝐷 are equal to vector 𝐯. 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 𝑥- and 𝑦-component. As this vector is moving right and down, it will have a positive 𝑥-component and a negative 𝑦-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 𝐀 has component 𝑥 one, 𝑦 one and vector 𝐁 has components 𝑥 two, 𝑦 two, then vector 𝐀𝐁 is equal to vector 𝐁 minus vector 𝐀. To calculate vector 𝐀𝐁, we can subtract the 𝑥- and 𝑦-components separately. Finally, we saw that the magnitude of vector 𝐯 with components 𝑥, 𝑦 is equal to the square root of 𝑥 squared plus 𝑦 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.