Rectangular Coordinates in 3-Space http://www.math.drexel.edu/~dp399/index.html Math 200 Week 1- Monday Rectangular Coordinates in 3-Space

Main Questions for Today Math 200 Main Questions for Today How do we describe where points are in 3-Space? How do we compute distances in 3-Space? Can we use the distance formula to write down equations of spheres? What are cylindrical surfaces?

Rectangular Coordinates in 2d Math 200 Rectangular Coordinates in 2d On the xy-plane, we describe where things are on the plane with an address of the form (x,y). (x,y) y x

Rectangular Coordinates in 3d Math 200 Rectangular Coordinates in 3d In 3D (3-Space) we lay the xy-plane down like a tabletop and add a z-axis vertically. z x y This is coming out of the screen toward us

Rectangular Coordinates in 3d Math 200 Rectangular Coordinates in 3d And we just add a z-component to our coordinates… (x,y,z) z z0 (x0,y0,z0) We want to describe where this point is in space x y y0 x0

There are actually two way to do this Math 200 There are actually two way to do this We’ll stick to the right-handed way, but you should be aware that there’s another way to draw the axes. *Just remember: with right-handed coordinates, the x-axis is coming towards us

Math 200 Coordinate Planes It’ll often be useful to refer to the three coordinate planes: xy-plane: where z=0 so all points look like (x,y,0) yz-plane: where x=0 so all points look like (0,y,z) xz-plane: where y=0 so all points look like (x,0,z)

Math 200 Distance in space We already know that a2 +b2 = c2 (Pythagorean Theorem) This gives us the distance formula in 2D (x2,y2) c b = y2 - y1 (x1,y1) a = x2 - x1

Pythagorean Theorem in 3D Math 200 Pythagorean Theorem in 3D x0 y0 x y z z0 Apply the Pyth. Thm. twice First use the 2D version to find c on the xy-plane Then use it again to find d. (x0,y0,z0) d c

Math 200 x0 y0 x y z z0 c d (x0,y0,z0)

Review Go to our homepage and click on the Monday Pt. 1 survey. Math 200 Review Go to our homepage and click on the Monday Pt. 1 survey. http://www.math.drexel.edu/~dp399/math200.html

Math 200 Spheres Let’s find an equation for a sphere with radius r with center (x0, y0, z0) A circle is the collection of points on a plane equidistant from a given center point. A sphere is the collection of points in space equidistant from a given center point. KEY IS DISTANCE THE FIXED DISTANCE WE CALL THE RADIUS

COMING UP WITH A GENERAL EQUATION Math 200 COMING UP WITH A GENERAL EQUATION We know that This is for a pair of fixed points. We just want the center to be fixed. Let’s replace (x2, y2, z2) with (x, y, z) and set the distance to r. We’ll square both sides to make it look more like the general equation for a circle: This is the general formula for a circle with radius r and center (x1, y1, z1)

Math 200 Example Consider a sphere whose diameter extends from A(2, -1, 0) to B(4, 3, 2) First we’ll find the midpoint of A and B which will be our center: Then we can find the radius: Answer:

A Little Theorem To see which it is, we have to complete the square. Math 200 A Little Theorem THEOREM: An equation of the following form represents a sphere, a single point, or has no graph. To see which it is, we have to complete the square. Consider the equation: x2 + y2 + z2 - 2x + 4y + 5 = 0 Complete the square for x, y, and z. What do you get?

Okay so we have a “radius” of 0… What does this mean? Math 200 Okay so we have a “radius” of 0… What does this mean? The only point that satisfies this equation is (1, -2, 0)

Math 200 Cylindrical Surfaces Spheres are a pretty easy first 3D shape to consider. Category of “easy” 3D shapes: cylindrical surfaces Not actually cylinders Equations missing one or more of our three variables E.g. y = x2: all points where y is the square of x and z can be anything at all x2 + z2 = 1: the x and z coordinates form a circle and y can be anything

Well, in 3D we have a z-component to worry about. Math 200 First consider y=x2 in 2D x -2 -1 1 2 y 4 Well, in 3D we have a z-component to worry about. If it’s not in the equation, then it’s free to take any value x -2 -1 1 2 y 4 z x -2 -1 1 2 y 4 z

Putting them all together… Math 200 Putting them all together… But really, there are infinitely many of these, so we get a sheet: