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Question Video: Finding the Components of the Sum of End-to-End Vectors on a Diagram Mathematics • 12th Grade

Shown on the grid of unit squares are the vectors ๐ฎ and ๐ฏ. What are the components of ๐ฎ? What are the components of ๐ฏ? What are the components of ๐ฎ + ๐ฏ?

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Video Transcript

Shown on the grid of unit squares are the vectors ๐ฎ and ๐ฏ. What are the components of vector ๐ฎ? What are the components of vector ๐ฏ? What are the components of vector ๐ฎ plus vector ๐ฏ?

In this question, weโ€™re given a diagram including two vectors, ๐ฎ and ๐ฏ. And itโ€™s important to note that the squares in this diagram are unit squares; their side length is one. We need to use this diagram to determine the components of vectors ๐ฎ, ๐ฏ, and ๐ฎ plus ๐ฏ. We can do this by first recalling that a vector is an object with magnitude and direction. We can represent this magnitude and direction in several different ways. Graphically, we can represent the direction by using an arrow and the magnitude by using the length of the arrow. For example, in the diagram, vector ๐ฎ is represented by an arrow. It has an initial point at the base of the arrow and a terminal point at the head of the arrow.

Itโ€™s important to note though vector ๐ฎ is entirely represented by its magnitude and direction. So we can represent this in terms of displacement. In particular, the horizontal and vertical displacements of these vectors are called the components of the vector. We can see that when moving from the initial point of vector ๐ฎ to the terminal point of vector ๐ฎ, we move three squares to the right. And since these are unit squares, we can say that we move three units to the right. Similarly, we can see that when we move from the initial point of vector ๐ฎ to the terminal point of vector ๐ฎ, we move two units up. And since we know that we call movement to the right the positive direction and movement upwards the positive direction, we can say the displacement of vector ๐ฎ is positive three units in the horizontal direction and positive two units in the vertical direction.

And we can represent this in vector notation. We need to use triangular brackets, where the first number will represent the horizontal displacement of the vector and the second number will represent the vertical displacement. Vector ๐ฎ has a horizontal displacement of three and a vertical displacement of two.

We can follow this exact same process to answer the second part of the question. The starting point of the arrow will give the initial point of vector ๐ฏ, and the head of the arrow will give the terminal point of vector ๐ฏ. And we see when we follow this arrow, we move two units to the right and we also move three units down. The horizontal displacement of vector ๐ฏ is positive two units in the horizontal direction and negative three units in the vertical direction. And once again, we can represent this in vector notation. The first component of our vector is two, and the second component of this vector will be negative three. ๐ฏ is the vector two, negative three.

In the final part of this question, we need to determine the components of the vector ๐ฎ added to the vector ๐ฏ. There are several different ways of doing this. However, since weโ€™re given a diagram, weโ€™re going to do this by using the given diagram. And we can do this by recalling one way of adding two vectors together is to sketch them tip to tail. We sketch the tip or head of the first vector to be the tail of the second vector. And we can see that this is already given in the diagram. The terminal point of vector ๐ฎ is coincident with the initial point of vector ๐ฏ.

We can then add the two vectors together by using the initial point of vector ๐ฎ as the initial point of vector ๐ฎ plus vector ๐ฏ and the terminal point of vector ๐ฏ as the terminal point of vector ๐ฎ plus vector ๐ฏ. This gives us the following sketch of vector ๐ฎ plus vector ๐ฏ. We can also think of this as the displacements of vectors ๐ฎ and ๐ฏ combined. So we can find the components of this vector by either combining the displacements from vectors ๐ฎ and ๐ฏ, finding the displacements of vectors ๐ฎ and ๐ฏ in the diagram, or just directly finding the components of vector ๐ฎ plus vector ๐ฏ from the diagram. And this is the method weโ€™ll use.

We see when we move from the initial point of vector ๐ฎ plus vector ๐ฏ to the terminal point of vector ๐ฎ plus vector ๐ฏ, we move five units to the right. So its horizontal displacement will be positive five. Similarly, when we move from the initial point of this vector to the terminal point of this vector, we move one unit downward. This means its vertical displacement is negative one. We can then write this in vector notation as the vector five, negative one.

And then itโ€™s worth pointing out we couldโ€™ve also done this by adding the horizontal displacements of vectors ๐ฎ and ๐ฏ and the vertical displacements of vectors ๐ฎ and ๐ฏ. Adding the horizontal displacements, we get three plus two is five. And adding the vertical displacements, we get two plus negative three is negative one. In either case, we were able to show ๐ฎ is the vector three, two; ๐ฏ is the vector two, negative three; and vector ๐ฎ plus vector ๐ฏ is the vector five, negative one.

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