O Level Math / High School – Transformations – vector, translation, reflection

hello in this session we will look at vectors and transformations in an earlier session I’ve talked about vectors and talked about some of the ways that you use them right and I’ve talked about how a vector how you might think of it as an arrow pointing in a certain direction and then it is basically a quantity that tells you the length and that tells you that the magnitude that means the size and the direction and it can be used to describe anything that has a magnitude and direction like the force or displacements or velocity and so on and then you can describe a vector by two numbers one number be the magnitude right for example you can say that the magnitude of this is five and five can be the force and 5 Newtons of distance like five meters and so on and a direction but in the direction you might describe that you you will need a reference direction for example to the north and this might be a certain angle by 20 degrees so you can say that the direction is so many degrees to the north so that’s the vector now in this session we are going to look at another way of describing this same nation to describe the same information about the direction and and the angle and the way to do that is to think of a graph before X and y-axis okay if I on the graph you can imagine drawing an arrow or a vector with to tell them at the origin the tip of the arrow would be somewhere on the graph if you draw them then there is another way to you can tell you can describe a vector you can make use of the position that arrow the position of the temple that arrow on the graph so let’s say for example the x value is 4 and the y value of the point is 3 so in a sense the 4 and 3 is an equally good description equally good as the magnitude and the distance that means that if I if instead of telling you the magnitude and the angle I tell you the X and the y value of the graph paper then you can actually draw the same vector simply by going to that point and drawing the knife pointing from 0 to that points so the information using x and y is an equally good way to represent or to describe the vector it contains the same information you can convert between X Y and length and angle alright so that’s another way to describe a vector and in our last session I’ve talked about matrix now we can simply write this information 4 and 3 and arrange them in the form of a matrix so one way to do that is to write 4 comma three so that’s like matrix with two columns and one row another way that is also of abused is by writing for and as three below that okay and that’s a matrix with two rows and one column now

we often call this a column matrix and in this session we go off we will mostly be looking at this way of writing the two numbers because the idea is then we can then make use of what we learnt in matrices the kind of calculations that we learned to do different things with with this vector in this point yeah so one thing that we could do is to see how we will add vectors using this ready to represent a vector with the x and y values can I search via a very easy way to add two vectors now let’s refresh our memory on how to add vectors suppose I have two vectors a and B okay but little if you remember I like to put a little squiggle below it the show that it’s not just a number but it’s actually a vector if I’ve to like this the way to end it there are two ways to add the vectors one is called a triangle law or condition one is called a parallelogram law so I’ve talked about both of these in the earlier session on vectors I’m going to make use of the well I’ll just talk about both right if we think about the parallelogram law I would just imagine that these two lines are the size of a parallelogram and I would complete the parallelogram at that by drawing the other two sides all right parallel all the sides and then this point up there would be the resultant the result of adding the two vectors so that’s how we do it and in the triangle law instead of drawing out the full parallelogram I would imagine taking this vector and shifting it until its tail touches the tip of the other vector so that meaning that you just draw one side of the turn one more side of the parallelogram and then you complete this triangle okay by drawing an additional line here and it’s really the same line and it gives you the same result ins okay so this is how you add vectors by essentially by drawing now using using this column matrix using this way of writing the vector with the two numbers and other acts of my values they have another way which is simply not to see how that works let me draw the axis the X and y axis right so suppose that I represent each of these vectors by the x and y values so let’s say this point here is 3-1 that point of that is 1/2 right so I have now the two sets of x and y values and represent the two vectors now if I look at the a the vector a think about what the three and the one means is the XY values that means that the x value means this that this point here is three drop a perpendicular from the tip of the vector to the x axis and if you go horizontally from the tip to the y axis it’s a 1 okay

now you can think about this getting from the origin to the tip of the arrow by moving three units from the origin three units to the right and then one unit up so one two three to the right and one unit up and you will reach the tip of a now my advice for the other arrow the tip is that the Y value of 2 and that’s value 1 going to from the origin to the tip of that arrow by going what do you mean to the right two units up right so how can I then get to that point which is the answer that we want well because this line is really the same as bad line same meaning same length and same direction they’re parallel now I can imagine starting from that point and go into that point by moving one unit to the right and two units up all right because that’s how I go from the origin to the tip of be all right when you lead to the right two units up so from the tip of a if I move one unit to the right and pull you this up I get to that point so one unit to the right and 2000 so what this really means is that in order to get here I have to start from this point which has my value of three and one so I must add one to this free to the next value I must add two to the Y value in order to get here so three all right something about three I move one unit to the right so three becomes four and I move two units out so one becomes two and this gives us a very simple way to actually find the answer you just see that what I’ve simply done is to add the X values of the two vectors to get four and the y values of the two vectors to get too many right both of these in column matrix form 3-1 so my vector if I want to find a plus B to get my answer which I I’ll call all its C there is something that time I can do it by writing a in this column vector column matrix or column vector form which is 3 1 and B which is 1 2 and I can simply add the x-values three plus one okay or the Y value 1 plus 2 to get 3 and that would be my resultant you can see that it’s now a simple all I need to do is these two simple additions and you don’t even have to draw the parallelogram of course you can only do it if you already have the X and the y values what if you don’t you have to find it you can find it by for example actually drawing it out on a piece of graph paper so that means you still need to draw something or you can find it using trigonometry alright because this is a triangle for example if you know that the line 5 and angle of some value then because this is a triangle you can then use trigonometry sines and cosines to find this side and that’s not now shall go into that at the moment but that’s addition using matrices next we’re going to look at

transformations let’s start on the graph paper and think about the points on the graph paper if what x value of 1 and a viper Y value of 2 now transformation is about moving points and shapes or figures around on the graph paper basically that’s the main thing that the main kind of transformation we look at at all level now and the ways that we can move this point around is there are a few ways that we will look at the first and simplest is called translation that means that if I take this point and move it to another point if I always do the same movement for any point that I have on the graph that’s called a translation now again if you are thinking of seeing this for the first time it may seem rather strange why you want to do this there are certain users especially when we have to do people do calculations in geometry and in designs and physics and engineering and and as well as many other areas of Suns and economics but what we are maybe when they look at is if you like a set of rules alright to tell us how you can do certain things on the graph paper and move the points and figures around this certain way so just treat it as a set of rules and like a little game if you like even though you might not my god might not like it it’s basically a set of rules I’ve already learned and this mu which is called a translation even one way that we can move points around on this XY plane so for example the translation is basically rule that tells us that we can move any points by a certain distance in a certain direction so for example if I take this point and the translation is that I can move it by this dispense okay say this is right to you list to the right and one units up and five another point for example day in couldn’t be I would do the same and so on that’s a transmission and to describe such translation we use what is called a translation vector given by these two and one two three we can write that matrix form but the idea is that if you are given any points on the graph for example if you’re doin B that B has a coop has coordinates of say minus 3 minus 4

okay so point B has these x and y values now we say that we can apply a translation to B and then we apply this translation be the point be moved from this point to that point and we can find the new points as I call it B right let us be with a stroke next to it I can find this point B Prime simply by adding the translation vector so B which is this vector goes to or is transformed to or you can even say it is mapped to if you’ve learned if you’ve seen the previous session on functions right B is mad to write it this way that’s that’s be is mapped to deep right which is given by B itself plus the translation vector here all we need is to add the two vectors and we get minus 4 plus 2 is minus 2 minus 3 plus 1 is minus 2 then we have our new points the point B they began after applying this translation to to B and in the same way if you have any other point like a or maybe C as long as you know the XY values you can just add the translation vector to get a new point that this is translation translation now moving on point it’s quite simple and maybe a little bit boring now what we could also do is to move a figure suppose that I have a triangle and I want to do a translation to it a translation vector of again to one home would I apply a translation to the whole figure the way is to find out the coding the the XY values of each point so let’s say this point a is one and one point B is 3 and say one and a half we see is two and three all right in order to find the new points the new position of this triangle we would add we will add to this back to my values code the translation vector itself so for a to find new put the new points you would take 1 1 and this will go to 1 1 + 2 1 when we add them you’ll get 3 + 2 + 4 point B in the same way it would take 3 1 x and y values and add to 1 so this

will go to 3 1 + 2 1 which is 5 2 and we will just repeat this for B not for C so once there so for each of this time we will get the x and y values for the new positions of a B and C and V and then we can put them in the graph so the new positions might be for example a my go to that may be there be Michael to that C and go to that and let me simply draw the new figure so we have a new figure and a translation we’re gonna look at reflection next suppose I have a point there with X Y of 1 & 2 now imagine that the x-axis is a mirror so we are looking at reflection in the next axis the x-axis is a mirror and I call this point a then a would have an image down here which are co-prime and we know that the image must be at the same distance behind the mirror so this is minus 2 would be the Y value of the image okay and you can imagine that if you have a whole figure like a triangle just now you can simply do a resection of the whole triangle by doing a reflection for each point on the triangle so just an example let me draw a triangle here it is C and B would be affected that C and I have a resection of the triangle now here as with translation we want to learn about how to find the coordinates the x and y values of of the point after the reflection so for example we have seen how to drill for a it’s quite simple you simply take the Y value and change it to a negative number and the one the one stays the same so it’s fairly straightforward now what we are going to do is to make use of matrices to help us to calculate the new position now it might seem be unnecessary because it is so simple just to change the Y value why bother to use matrices the reason is because later on we are going to look at slightly more complicated transformation where it is not so easy to find the new positions of the point so what we could do is to make use of this very simple example to see how we can actually apply matrices to the to define the new position now I’m just going to sketch out two right now the method first and then I will explain if I have a point a and x

and y runnings are given by one and two okay I want to find the position of a prank the the position after a is reflected in the x-axis and the way to do it using matrices is to multiply this 1 to this column vector by a matrix and that matrix is 1 minus 1 1 0 now when you multiply this matrix you take to get the first number you multiply the first row by the first column you multiply the number by the corresponding number so 1 times 1 which is 1 minus 1 times 2 which is minus 2 and you do that for the second number second row and first column 2 0 times 1 which is 0 1 times 2 which is 2 and the answer is my is right it looks like I’ve done something wrong here I’ve got the wrong matrix correct matrix should be 10.1 that’s the correct matrix now let me multiply that out again to get the first number look at the first row and there’s a second first column so 1 times 1 gives you 1 0 times 2 is 0 to get the next number you take the second row and the first column 0 times 1 is 0 minus 1 times 2 is minus 2 so you get 1 minus 2 1 minus 2 which is the correct position a prime and the idea is that you can apply that to any points it at this point P for example let’s say that’s 3 & 2 point 5 3 is 3 & 2 point 5 I can find a new position B Prime if I multiply the vector P by the same matrix 1 0 minus 1 0 and you’ll find that the answer when you carry out the multiplication is 3 and minus 2.5 and that gives the position of P so that’s the idea of using matrix now why does it work the reason why it works is that we know in this example that we really just need to change the Y value to negative just to change the sign the Y value so if it’s negative you change it to a positive for example and this matrix does them now you can see that way when I won’t do the multiplication I multiply the act these are X Y values and multiply the x value by the one here the Y value is just multiplied by zero when I try to get the first value so when I get the new value of x it always ends up giving the same value as the original expanding because it’s just multiplied by the one right together the new X money and for the Y value I get it by multiplying the first row first column but the first number in the back there’s always x zero please I can ignore that and Y value is always x our minus one which is why the sign is changed so that’s how testin effect that the matrix has on on the X of my values this particular matrix the effect of this particular matrix will multiply it’s only to change the sign of

the Y value so that’s why if you apply it to any other column vectors you get the same effect the Y value has the sign changed the ads remain the same you’ll stop here for this session see you next time