Understanding Coordinate Systems and Angles

Good day in this video. I would like to as a further introductory video quickly introduce corners systems and radiance Before we get into that here's a thought for the video Science is a way of thinking much more than a body of knowledge as a quote by Carl Sagan a very famous theoretical physicist Then I couldn't agree with that more. It's not but how what you know is think it's more about how you use it's also a lot about how you think about things okay so the goal is is this video is to briefly introduce a difference between the Cartesian and the polar coordinate systems as well as given introduction to degrees and radians as a measure of the size of an angle so firstly just a general overview of coordinate systems so coordinate systems help us to describe or determine the position of a point in space. It generally consists of an origin which we label O, so that's an O it's not a zero, and the point is the position is given in reference to O with the help of some of some axes at specific angles or direction with some units or labels as well as a corner system consists of instructions on how to label the point. relative to the origin and the axis. So we choose this fixed reference point which we call the origin and then the coordinate systems help us determine the position of a point in reference to that. The one system you should all be familiar with is the Cartesian coordinate system which consists out of the X and Y axis. The Y axis is typically in the vertical and yeah, let's just first say that these axes are a 90 degree angle towards the other. The Y axis is in the vertical. This is the direction of increasing values on the Y axis. And if you go down below the origin, you go into the negative values on the Y axis. And the X axis in the horizontal. So like I said, they are at right angles with regards to each other. And the X axis increase. The positive direction for the X axis is to the right. And the negative direction is to the left. left so in general we label any point by X coordinates X and Y so that's a general point so what does the X and the Y mean these coordinates well in reference to the origin they actually indicate the distance that you are from the origin along the specific axes so the X value here indicate actually the length of this line which is parallel to the x-axis from the origin and the y-coordinate indicates the length of this line which is also parallel to the y-axis. So the coordinates basically tell you how far in each direction along the x-axis and how far along the y-axis are you from the origin in terms of this point. relative to the origin. You can also, sorry to go back, you can also indicate this point with a line from an arrow from the origin pointing towards the line. And typically, we call this line R. And since in relation X and the relation of X and Y to R is, okay, so R is at some angle, if you measure it from the X axis, some angle theta. this lines is not lying on the x-axis that's pointing to this point x and y and the relation of r with regards to x and y is because the cortesian coordinates are a right angle that means that this also forms a right angle so x and y are the two side lengths of the right triangle whereas r is the hypotenuse of the right of the triangle this fact we will use a lot in terms of coordinate systems so if we go to the polar coordinate system if we want to describe exactly the same point this point which we had previously in terms of X and Y in terms of the polar coordinate system this will indicate be indicated by R and theta so how does that work so in terms of the polar coordinate system we still have an origin O as well as a reference line in the horizontal. but this is at the angle theta equals to zero degrees and we measure our angle in reference to this horizontal reference line and this point then is indicated by r theta in terms of its unique coordinates in the polar coordinate system so how does the polar coordinate system work well it's an it's an origin as well as a reference line of theta equal to zero we measure the angle theta counterclockwise direction so that means from the theta equal to zero we measure it in this direction so counterclockwise and r indicates a distance that you are from the origin and for this specific point and that will uniquely define this point why do we need both r and theta well if you So let's look at this point as an example at a fixed distance from what if we say We've got a point in a specific fixed distance The only thing that that defines if we just are at a fixed distance is that you? Correspond you on some circle with that centered on the origin So in this case we can look if we look at the intercepts with the axes We see actually that the length of this line the radius of the circle is 7 Now, if you were to ask me what is this point this arrow is pointing to in terms of polar coordinates, well, you know it's a distance of 7 from the origin, but that doesn't uniquely define the point because all of these points that form the circle are also a distance of 7 from the origin. So, in addition, you need to specify the angle. So, let's take a number example. So, we are looking at... this point that this arrow is pointing to, and it is at a distance of three from the origin. But that means if you're just looking at a distance of three from the origin, it could be any point on the circle. So once again, we need to specify the angle. So you need to specify both R and theta, two things. You can't just specify R. Okay. So these define these unique points. So how do we describe them in terms of the polar coordinate system? Well, we've already established that the radius of this line is 7. The length, I'm talking about the blue. So we want to illustrate that point over there. Now it is a distance of r is equal to 7 because the radius of the circle that's centered on the origin that goes through this point is 7. And we can see that by just looking at the intercept with the reference line. So we know the length of... r in this case will be equal to 7 so we don't need the circle anymore what else can we say well we need the angle theta and in this case the angle theta will actually be 45 degrees how do we know this well if you remember your special triangles the point that we had was originally this point this is exactly the same point that we were describing and you'll see that if you look at this right if you look at this triangle. This side length is 5. This side length is also 5, which means, well, and this side length is almost 7. Not quite like it. It will be within error. It'll be 7 because it's the square root of 50, so it's, and 7 squared is 49 if you use Pythagoras, but you can see that it's pretty much a 45 degree angle so actually this point is not exactly fine but it's somewhere on the fire so just from looking at special triangles if you have an equilateral triangle like this with the same side length at a 90 degree angle we know that these two angles must be 45 degrees so they sum up to 190. so at this point the one mark marked by the arrow here in the polar corner system will be given by seven comma 45 degrees which is the r of this point and the theta of this point now what about this point the one line on the reference line well we know the distance from the axis in this case is clear that it is freeze three since it's lying on the origin so we can say that r in this case will be equal to three now what would be the angle well it's lying we measure our angle in reference to the reference line in this case is lying on the reference line so it should be clear that this point in the polar corn system will be labeled by three comma at zero degrees so once again for this point your r is equal to three and your angle are at zero degrees which means it lies on the x-axis. Incidentally, if the point was over here, also at a distance of 3, what would be the angle? Think about it for a second. In this case, this point will be given by the polar coordinates 3 and 180 degrees, because from the reference angle, that point that is 180 degrees so i hope you all would agree with that okay so to compare the cartesian and the polar coordinates or just is or two ways to describe exactly the same point and it becomes useful especially polar coordinates are especially useful if you look at rotation because typically something is rotating around a fixed point so If you know the distance from that fixed point where you can put the origin, you just need to know the angle to describe its position uniquely. But when we get to rotational motion, I will cover these in detail again. So once again, the cortes and the polychord systems are just two different ways of describing a point in space in reference to some origin. So in this case, in terms of the relation, since these describe exactly the same point, I hope you would agree that if you are given the x and y coordinates, that you can write r and theta are given by this relationship, whereas r is just this length, which from Pythagoras'theorem is just y squared. the side length squares and some of them pull the root and as well as the angle is just inverse tan of this the size of this angle is given by the inverse tan relationship of y over x as i've indicated in the previous some of the previous slides this distance exactly x because our cartesian coordinate axes are at a right angle so we form this right angle triangle So it's up to you to now tell me if I'm given R and theta what would be the relationship to X and Y in the corner system? So what's the corresponding relationship of X? What's the function of X in terms of R and theta and Y in terms of R and theta and if you're just thinking about your Known trigonometry, it should be pretty straightforward What these are? but that's for you to do okay let's just quickly talk about the radian so the radian is it's another way of measuring the size of an angle so and it's defined in terms of also um in terms of an angle and distance from the origin so the angle is described by if you're looking at this circle how far along the arc of the circle you've traveled so in this case s indicates the arc length and that's this fraction of the circle that is spanned by this angle theta so the angle is given by the ratio of the arc length so how much of the circle is spanned by this angle divided by the radius of the circle so in terms of if you have the full circle then the circumference of a full circle is two pi radius spanned by the radius so a full so from this you can gather that a full circle is given by two radians because the R's cancel out. So the full circle is the angle describing a full circle which in degrees is 360 degrees the equivalent radians is 2 pi radians. Now what is the SI unit of radians? Well we describe it as radians but you would agree that the radius if from this definition the arc length along the arc. That is a distance and the radius is also a distance. So it's got length over length. So it's actually dimensionless. so it's a number we just describe that as radians so we just indicate this angle so if you give me an angle you have to say it's one radian or how many ever radians so that's the unit but it's dimensionless in terms of there's no specific si units for this angle because it's a length divided by length if you're not sure what i'm talking about please go watch the video about units and dimensions so comparing degrees to radians so one radians is typically much larger than one degree so if you look at the comparison so the full circle here in terms of comparing degree that's the full circle in terms of degrees that's the full circle in terms of radians so if you're looking comparatively divided two by the other that's a number with no 360 divided by 2 pi and that will give you that one radian is 57.3 degrees so Here's a formula to convert from radians to degrees, but the best is just to remember that 2 pi radians is equal to 360 degrees. And that will also help you with the method that I introduced in, I think, also the units and I mentioned video in terms of describing these angles or conversion between angles. OK. The other thing you need to note about degrees and radians is that your calculator can be set to either accept degrees or radians. So please make sure that it's set to the unit that you are inputting to the angle to or to the calculator to calculate the angles. For example if you take the inverse turn of this number of this ratio between y over x you will get an angle of 30 degrees degrees but if you put in the same number you will get an answer of 0.365 now the calculator doesn't tell you whether it's giving it to you in degrees or radians it just gives you the number so if you're not careful so if you input something and you think you're working in degrees and you're actually in radians you're going to get the wrong answer just make sure that your calculator is set to the appropriate angular units for the problem so that you are inputting. So do you need to convert degrees to radians or vice versa? And we could use any of these angular units in physics 101. So just to recap, I've introduced the polar and cortesian coordinates. Hopefully it's clear now what the relationship are between these two coordinate systems as well as a brief interview of degrees. or a brief introduction to degrees and radiance feel free to put your questions in the comment section or but other than that i'll see you in the next video

It generally consists of an origin which we label O, so that's an O it's not a zero, and the point is the position is given in reference to O with the help of some of some axes at specific angles or direction with some units or labels as well as a corner system consists of instructions on how to label the point. relative to the origin and the axis. So we choose this fixed reference point which we call the origin and then the coordinate systems help us determine the position of a point in reference to that.

The one system you should all be familiar with is the Cartesian coordinate system which consists out of the X and Y axis. The Y axis is typically in the vertical and yeah, let's just first say that these axes are a 90 degree angle towards the other. The Y axis is in the vertical.

This is the direction of increasing values on the Y axis. And if you go down below the origin, you go into the negative values on the Y axis. And the X axis in the horizontal.

So like I said, they are at right angles with regards to each other. And the X axis increase. The positive direction for the X axis is to the right. And the negative direction is to the left.

left so in general we label any point by X coordinates X and Y so that's a general point so what does the X and the Y mean these coordinates well in reference to the origin they actually indicate the distance that you are from the origin along the specific axes so the X value here indicate actually the length of this line which is parallel to the x-axis from the origin and the y-coordinate indicates the length of this line which is also parallel to the y-axis. So the coordinates basically tell you how far in each direction along the x-axis and how far along the y-axis are you from the origin in terms of this point. relative to the origin. You can also, sorry to go back, you can also indicate this point with a line from an arrow from the origin pointing towards the line.

And typically, we call this line R. And since in relation X and the relation of X and Y to R is, okay, so R is at some angle, if you measure it from the X axis, some angle theta. this lines is not lying on the x-axis that's pointing to this point x and y and the relation of r with regards to x and y is because the cortesian coordinates are a right angle that means that this also forms a right angle so x and y are the two side lengths of the right triangle whereas r is the hypotenuse of the right of the triangle this fact we will use a lot in terms of coordinate systems so if we go to the polar coordinate system if we want to describe exactly the same point this point which we had previously in terms of X and Y in terms of the polar coordinate system this will indicate be indicated by R and theta so how does that work so in terms of the polar coordinate system we still have an origin O as well as a reference line in the horizontal. but this is at the angle theta equals to zero degrees and we measure our angle in reference to this horizontal reference line and this point then is indicated by r theta in terms of its unique coordinates in the polar coordinate system so how does the polar coordinate system work well it's an it's an origin as well as a reference line of theta equal to zero we measure the angle theta counterclockwise direction so that means from the theta equal to zero we measure it in this direction so counterclockwise and r indicates a distance that you are from the origin and for this specific point and that will uniquely define this point why do we need both r and theta well if you So let's look at this point as an example at a fixed distance from what if we say We've got a point in a specific fixed distance The only thing that that defines if we just are at a fixed distance is that you? Correspond you on some circle with that centered on the origin So in this case we can look if we look at the intercepts with the axes We see actually that the length of this line the radius of the circle is 7 Now, if you were to ask me what is this point this arrow is pointing to in terms of polar coordinates, well, you know it's a distance of 7 from the origin, but that doesn't uniquely define the point because all of these points that form the circle are also a distance of 7 from the origin.

So, in addition, you need to specify the angle. So, let's take a number example. So, we are looking at... this point that this arrow is pointing to, and it is at a distance of three from the origin.

But that means if you're just looking at a distance of three from the origin, it could be any point on the circle. So once again, we need to specify the angle. So you need to specify both R and theta, two things.

You can't just specify R. Okay. So these define these unique points. So how do we describe them in terms of the polar coordinate system?

Well, we've already established that the radius of this line is 7. The length, I'm talking about the blue. So we want to illustrate that point over there. Now it is a distance of r is equal to 7 because the radius of the circle that's centered on the origin that goes through this point is 7. And we can see that by just looking at the intercept with the reference line. So we know the length of...

r in this case will be equal to 7 so we don't need the circle anymore what else can we say well we need the angle theta and in this case the angle theta will actually be 45 degrees how do we know this well if you remember your special triangles the point that we had was originally this point this is exactly the same point that we were describing and you'll see that if you look at this right if you look at this triangle. This side length is 5. This side length is also 5, which means, well, and this side length is almost 7. Not quite like it. It will be within error.

It'll be 7 because it's the square root of 50, so it's, and 7 squared is 49 if you use Pythagoras, but you can see that it's pretty much a 45 degree angle so actually this point is not exactly fine but it's somewhere on the fire so just from looking at special triangles if you have an equilateral triangle like this with the same side length at a 90 degree angle we know that these two angles must be 45 degrees so they sum up to 190. so at this point the one mark marked by the arrow here in the polar corner system will be given by seven comma 45 degrees which is the r of this point and the theta of this point now what about this point the one line on the reference line well we know the distance from the axis in this case is clear that it is freeze three since it's lying on the origin so we can say that r in this case will be equal to three now what would be the angle well it's lying we measure our angle in reference to the reference line in this case is lying on the reference line so it should be clear that this point in the polar corn system will be labeled by three comma at zero degrees so once again for this point your r is equal to three and your angle are at zero degrees which means it lies on the x-axis. Incidentally, if the point was over here, also at a distance of 3, what would be the angle? Think about it for a second. In this case, this point will be given by the polar coordinates 3 and 180 degrees, because from the reference angle, that point that is 180 degrees so i hope you all would agree with that okay so to compare the cartesian and the polar coordinates or just is or two ways to describe exactly the same point and it becomes useful especially polar coordinates are especially useful if you look at rotation because typically something is rotating around a fixed point so If you know the distance from that fixed point where you can put the origin, you just need to know the angle to describe its position uniquely.

But when we get to rotational motion, I will cover these in detail again. So once again, the cortes and the polychord systems are just two different ways of describing a point in space in reference to some origin. So in this case, in terms of the relation, since these describe exactly the same point, I hope you would agree that if you are given the x and y coordinates, that you can write r and theta are given by this relationship, whereas r is just this length, which from Pythagoras'theorem is just y squared. the side length squares and some of them pull the root and as well as the angle is just inverse tan of this the size of this angle is given by the inverse tan relationship of y over x as i've indicated in the previous some of the previous slides this distance exactly x because our cartesian coordinate axes are at a right angle so we form this right angle triangle So it's up to you to now tell me if I'm given R and theta what would be the relationship to X and Y in the corner system?

So what's the corresponding relationship of X? What's the function of X in terms of R and theta and Y in terms of R and theta and if you're just thinking about your Known trigonometry, it should be pretty straightforward What these are? but that's for you to do okay let's just quickly talk about the radian so the radian is it's another way of measuring the size of an angle so and it's defined in terms of also um in terms of an angle and distance from the origin so the angle is described by if you're looking at this circle how far along the arc of the circle you've traveled so in this case s indicates the arc length and that's this fraction of the circle that is spanned by this angle theta so the angle is given by the ratio of the arc length so how much of the circle is spanned by this angle divided by the radius of the circle so in terms of if you have the full circle then the circumference of a full circle is two pi radius spanned by the radius so a full so from this you can gather that a full circle is given by two radians because the R's cancel out.

So the full circle is the angle describing a full circle which in degrees is 360 degrees the equivalent radians is 2 pi radians. Now what is the SI unit of radians? Well we describe it as radians but you would agree that the radius if from this definition the arc length along the arc.

That is a distance and the radius is also a distance. So it's got length over length. So it's actually dimensionless. so it's a number we just describe that as radians so we just indicate this angle so if you give me an angle you have to say it's one radian or how many ever radians so that's the unit but it's dimensionless in terms of there's no specific si units for this angle because it's a length divided by length if you're not sure what i'm talking about please go watch the video about units and dimensions so comparing degrees to radians so one radians is typically much larger than one degree so if you look at the comparison so the full circle here in terms of comparing degree that's the full circle in terms of degrees that's the full circle in terms of radians so if you're looking comparatively divided two by the other that's a number with no 360 divided by 2 pi and that will give you that one radian is 57.3 degrees so Here's a formula to convert from radians to degrees, but the best is just to remember that 2 pi radians is equal to 360 degrees.

And that will also help you with the method that I introduced in, I think, also the units and I mentioned video in terms of describing these angles or conversion between angles. OK. The other thing you need to note about degrees and radians is that your calculator can be set to either accept degrees or radians.

So please make sure that it's set to the unit that you are inputting to the angle to or to the calculator to calculate the angles. For example if you take the inverse turn of this number of this ratio between y over x you will get an angle of 30 degrees degrees but if you put in the same number you will get an answer of 0.365 now the calculator doesn't tell you whether it's giving it to you in degrees or radians it just gives you the number so if you're not careful so if you input something and you think you're working in degrees and you're actually in radians you're going to get the wrong answer just make sure that your calculator is set to the appropriate angular units for the problem so that you are inputting. So do you need to convert degrees to radians or vice versa? And we could use any of these angular units in physics 101. So just to recap, I've introduced the polar and cortesian coordinates. Hopefully it's clear now what the relationship are between these two coordinate systems as well as a brief interview of degrees.

or a brief introduction to degrees and radiance feel free to put your questions in the comment section or but other than that i'll see you in the next video

Transcript for:Understanding Coordinate Systems and Angles

Transcript for:
Understanding Coordinate Systems and Angles