WEBVTT
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Shown on the grid of unit squares are the vectors ๐ฎ, ๐ฏ, and ๐ฎ plus ๐ฏ.
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What are the components of ๐ฎ?
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What are the components of ๐ฏ?
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What are the components of ๐ฎ plus ๐ฏ?
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In this question, weโre given a diagram of three vectors: the vector ๐ฎ, the vector ๐ฏ, and the vector ๐ฎ plus ๐ฏ.
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However, instead of our vectors being given with an axes in the origin, instead weโre just given them on a diagram of unit squares.
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We need to use this grid of unit squares to determine the components of ๐ฎ, the components of ๐ฏ, and the components of ๐ฎ plus ๐ฏ.
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To answer this question, weโre first going to need to recall exactly what we mean by the components of a vector.
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Now itโs important to remember that vectors are an object with magnitude and direction.
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They donโt necessarily represent movement.
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They can represent a lot of different things.
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However, we can always represent vectors graphically by choosing a coordinate system.
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And it can be very useful to think of them in this way.
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When represented graphically, the vector ๐, ๐ can represent movement ๐ units horizontally and ๐ units vertically.
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Or, if you prefer, the horizontal change will be ๐ and the vertical change will be ๐.
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So what does this mean for the three vectors weโve shown on our grid of unit squares?
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Well, first, because each of the squares in our grid has unit length, we can just count the number of squares to determine the horizontal change and vertical change in each of our vectors.
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We can then use this to find the components of our vector because the components of our vector are the values of ๐ and ๐, the horizontal change and the vertical change.
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So letโs start with our vector ๐ฎ.
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We need to determine its horizontal change and its vertical change.
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On our standard coordinate axes, as we move to the right, we increase our horizontal value, and as we move upward, we increase our vertical value.
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So we can see when we start at the initial point of our vector ๐ฎ and end at the terminal point of our vector ๐ฎ, we move two units to the right and one unit up.
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So weโve increased our horizontal value by two and weโve increased our vertical value by one.
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So the horizontal component is two and the vertical component of vector ๐ฎ is one.
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Therefore, ๐ฎ is the vector two, one.
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We can then do exactly the same for vector ๐ฏ.
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This time, when we start at the initial point of vector ๐ฏ and end at the terminal point of vector ๐ฏ, we can see we move down four units and to the left three units.
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Since we move down four units, weโve decreased our vertical value by four.
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In other words, our vertical component will be negative four.
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And because weโve moved to the left three units, weโve decreased our horizontal value by three.
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So the horizontal component of our vector is negative three.
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Therefore, ๐ฏ is the vector negative three, negative four.
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Now, all we need to do is exactly the same for our vector ๐ฎ plus ๐ฏ.
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Letโs see the change in our horizontal position and vertical position when we move from the initial point of our vector ๐ฎ plus ๐ฏ to the terminal point in our vector ๐ฎ plus ๐ฏ.
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We see we move one unit to the left and three units down.
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So the horizontal change is negative one and the vertical change is negative three.
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This gives us that the vector ๐ฎ plus ๐ฏ is the vector negative one, negative three.
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And we could stop here; however, there is one thing worth pointing out about our vector ๐ฎ plus ๐ฏ.
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This isnโt the only way we couldโve found this vector.
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We couldโve also added our two previous vectors together because, remember, when we add two vectors together, we can do this graphically by drawing one vector after the other so the terminal point of our first vector is the initial point of our second vector just like weโve done in this picture.
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Then moving along our two vectors is the same as adding these two vectors together.
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However, thereโs a second way of adding these two vectors together, where we add their components together.
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And all this really means is weโre adding the horizontal and vertical components of each vector separately.
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So letโs use this method to check our answer.
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The vector ๐ฎ plus the vector ๐ฏ will be the vector two, one plus the vector negative three, negative four.
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We need to add the corresponding components together.
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So letโs start by adding the first two components of our vectors together.
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So the first component is going to be two plus negative three.
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And itโs worth reiterating here what this means is if we travel along vector ๐ฎ and vector ๐ฏ, first we travel along vector ๐ฎ, so we change the horizontal value by two.
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And then we travel along vector ๐ฏ, so we change the value by negative three.
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We can then do exactly the same thing in the vertical direction or for our second components.
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We add the second component of each vector together to get one plus negative four.
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And if we calculate both of these components, we get the vector negative one, negative three, just as we did before.
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Therefore, given the sketch of the vectors ๐ฎ, ๐ฏ, and ๐ฎ plus ๐ฏ on a grid of unit squares, we were able to find the components of ๐ฎ, the components of ๐ฏ, and the components of ๐ฎ plus ๐ฏ.
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We showed that ๐ฎ was the vector two, one; ๐ฏ was the vector negative three, negative four; and ๐ฎ plus ๐ฏ was the vector negative one, negative three.