In the first half of this video, we briefly introduce X and Y axes that form the Cartesian x y plane, and discuss coordinates of points with respect to the axis. The Cartesian plane is an incredibly useful device for using mathematics, to analyze phenomena that occur in two-dimensional spacial settings. We go further in the second half of this video, and introduce a third Z axis, that allows us to model phenomena in three-dimensional spacial settings. You might have heard the expression thinking outside the square, or thinking outside the box. Well, maybe the term Lateral Thinking, originally due to Edward de Bono. All of which are synonymous with creative, or novel ways of thinking, or viewing ideas of problems, utilizing a completely new or different perspective. The 17th century mathematician philosopher Rene Descartes, was one of the most original thinkers in the history of mathematics. He literally thought laterally in many ways, and in particular to discover his remarkable invention the Cartesian plane the topic of this video. In Latin his name was Renatus Cartesius from which the adjective Cartesian is derived. We've discussed the real line in previous videos. Descartes imagined not just one, but two real lines positioned in space, so that they intersect at zero and are mutually perpendicular. What could be more lateral than that? The real line from left to right he called the x-axis. The real line up and down he called the y-axis. Think of the x-axis as horizontal, and the y-axis as vertical. Notice how the axes are perpendicular to each other and the intersection point is called the origin, which we denote by O. O for origin and O also for zero. Now, this is a special case of the following; take any point at all on this diagram, and call this point p and move down to the closest point on the x-axis. Say a, and move across to the closest point on the y-axis, say b, then the pair a,b forms what's called the coordinates of P. If you imagine this diagram extending infinitely to the right, to the left, and up and down, you get what's called the Cartesian plane or the x,y-plane. Let's practice with some real points. So, here's a copy of the x,y-plane with some points marked on the axes, and here are three points that we want to plot, P, Q and R. The point P has coordinates (2,1), point Q has coordinates (6,4), and the point R has coordinates (6,1). You can see that they form the vertices of a right-angled triangle. You could ask for example for the length of the hypotenuse, h say, and then use the theorem of Pythagoras. Remember, the square of the hypotenuse is the sum of the squares of the other sides of the right-angled triangle. As we move from P to R, we go from two to six on the x-axis, that's four units. As we go from R to Q, we move from one to four on the y-axis, that's three units. So, the sum of the squares of the two sides of the triangle are four squared plus three squared, which is five squared. So, h is equal to five. This calculation is a special case of a more general result. If we have points P and Q in the x,y-plane, then the distance D between them is given by this formula. D is just the square root of the sum of the squares of the differences of the coordinates. The formula for D comes about as before by finding a right-angled triangle in the plane, and using the theorem of Pythagoras. Now, we don't live in a flat world, we live in three dimensions. So, you could ask, can we adapt Descartes's lateral thinking to include a third copy of the real line? Yes. We can think of the x,y-plane now as horizontal like a table top in your kitchen, or your dining room. Then, we'd taken a new copy of the real line, which we call the z-axis, that emanates out of the table top and moves up and down vertically. Again, we denote the origin by big O, now we have three zeros. So, how do we model points in space? Here are the axes again, and I've got some point P suspended in space. Imagine it's sort of hovering over the table top, which is the x,y-plane. We dropped down from P until we hit the x,y-plane, at say the point Q. We move across to the x and y axis at the points say a and b, and we can move directly across horizontally to the z-axis, at some points say c. These numbers a,b and c form what's called the coordinates of P. Notice that the coordinates of Q are a,b and zero. Then, we can ask for example, What's the distance from P to the origin O? Call this distance d, say. Can you say Pythagoras working? The triangle OPQ is in fact a right-angle triangle. With hypotenuse d, one of the shortest side lengths is this vertical length c, the horizontal length will call h say. But h is itself a hypotenuse of another right-angled triangle, formed by O, Q and the point A on the x-axis. The shortest side lengths in this triangle are a and b. So, if we apply Pythagoras to that triangle that I've shaded gray that's lying horizontally, we get that h squared is a squared plus b squared. But if we apply Pythagoras to the other triangle, we see that d squared is h squared plus c squared, which is just a squared plus b squared plus c squared. So, we conclude that d is the square root of the sum of the squares of the coordinates of P. Thus, we have the following result. That the distance from a point in space to the origin is just the square root of the sum of the squares of the coordinates. This works regardless of whether the coordinates are positive, negative, or zero. In the diagram I had everything, so the coordinates are positive but the mathematics still works. Let's look at an example. Do you remember the unit square, with side lengths one and diagonal the square root of two? What if we were to consider a unit cube, where the side lengths are all one? Here's an example of a cube, and we can imagine all the side lengths are one, and then we can ask, what's the length of the diagonal of the cube, that is the distance from one corner to the opposite corner? Of course you can figure it out directly, but let's put it in the context of our discussion of coordinates of points in space. Let's position the cube, so that the point furthers the way is at the origin, and then the point closest to us has coordinates (1,1,1), because it's the unit cube. Call that point P, and then by our formula the distance from P to the origin is just the square root of the sum of the coordinates. If we draw in the diagonal d, then d is the square root of three. The theorem of Pythagoras is ubiquitous in mathematics. Because it describes the length of the hypotenuse as the square root of the sum of two numbers, you expect to say square root signs a lot. Remember, expressions using square root signs are called Surds. So, we'll see a lot of surd expressions in this course, and indeed in any course on calculus. We've covered a lot in just a few minutes. Please read and digest the notes accompanying this video. When you're ready please attempt the exercises, and we look forward to seeing you again soon.
