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