Rectangular Coordinates in Space Philippe B. Laval KSU Today Philippe B. Laval (KSU) Rectangular Coordinates in Space Today 1 / 11
Introduction We quickly review one and two-dimensional spaces and then present the three-dimensional space. Philippe B. Laval (KSU) Rectangular Coordinates in Space Today 2 / 11
Review: One-dimensional Space The one-dimensional space is also known as R. Its geometric representation is a line. To locate points on a line, only one number is needed. That number is called the coordinate of the point. It simply represents the distance between a fixed origin usually denoted O and the point. By convention, the origin is assigned the coordinate 0. Positive coordinates are to the right of the origin, negative coordinates are to the left of the origin. If P and Q are two points with respective coordinates x 1 and x 2, then the distance between P and Q, denoted PQ is PQ = x 2 x 1 (1) = (x 2 x 1 ) 2 Philippe B. Laval (KSU) Rectangular Coordinates in Space Today 3 / 11
Review: Two-dimensional Space The two dimensional-space is known as R 2 also denoted R R. Its geometric representation is a plane. To locate points, we need two perpendicular axes, called the coordinate axes. By convention, the x-axis is horizontal and the y-axis is vertical. The point where they intersect is the origin which is assigned the coordinate (0, 0). Every point in the plane can be represented as an ordered pair (x, y) where x is the x-coordinate which represents how far the point is from the y-axis and y is the y-coordinate which represent how far the point is from the x-axis. Thus, R 2 = R R = {(x, y) : x R, y R} Philippe B. Laval (KSU) Rectangular Coordinates in Space Today 4 / 11
Review: Two-dimensional Space If we have two points P (x 1, y 1 ) and Q (x 2, y 2 ), then using the Pythagorean theorem, we have PQ 2 = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 Therefore PQ = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 (2) Given two points P (1, 3) and Q (2, 4), find PQ A circle in the plane is defined to be the set of points at a fixed distance (called the radius) of a given point (called the center of the circle). What is the equation of the circle of radius r > 0, centered at the point P of coordinates (h, k)? Philippe B. Laval (KSU) Rectangular Coordinates in Space Today 5 / 11
Three-dimensional Space The 3-D space, is denoted R 3 also known as R R R. Its geometric representation is space. In order to represent points in space (3-D), we first select a fixed point we call the origin. We then select three directed lines, perpendicular to each other and going through the origin. These axes are called coordinate axes. We must adopt some convention to determine the positive and negatives directions for each axes. Usually, in mathematics, the x and y-axes are horizontal and the z-axis is vertical. Their direction being determined by the right-hand rule. In addition to the coordinate axes, three planes play an important role. They are the planes containing two of the coordinate axes, they are called the coordinate planes. There are three of them. They are the xy-plane, the yz-plane and the xz-plane. Philippe B. Laval (KSU) Rectangular Coordinates in Space Today 6 / 11
Three-dimensional Space Figure: Right-handed Coordinate System Philippe B. Laval (KSU) Rectangular Coordinates in Space Today 7 / 11
Three-dimensional Space If a point P has coordinates (a, b, c) then a represents the distance between the point and the yz-plane, b represents the distance between the point and the xz-plane and c the distance between the point and the xy-plane. On the xy-plane, we always have z = 0. This is the equation of the xy-plane. Similarly, the equation of the xz-plane is y = 0 and the equation of the yz-plane is x = 0. If P (x 1, y 1, z 1 ) and Q (x 2, y 2, z 2 ) are two points in space, then the distance between them is give by PQ = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 + (z 2 z 1 ) 2 A sphere is defined to be the set of points at a fixed distance (called the radius) of a given point (called the center of the sphere). What is the equation of the sphere of radius r > 0, centered at the point P of coordinates (h, k, l)? Philippe B. Laval (KSU) Rectangular Coordinates in Space Today 8 / 11
Basic Geometric Shapes Here, we look at the equations of some known shapes in the plane and see what objects the same equation will generate in space. Describe the set of points in the plane given by x = 3. Do the same in space. Describe the set of points in the plane given by y = 2x + 3. Do the same in space. Describe the set of points in the plane given by z = 2x + 3. Do the same in space. Philippe B. Laval (KSU) Rectangular Coordinates in Space Today 9 / 11
Basic Geometric Shapes Describe the set of points in the plane given by x 2 + y 2 = 4. Do the same in space. Describe the set of points in the plane given by x 2 + z 2 = 4. Do the same in space. Describe the set of points in the plane given by x 2 + y 2 4. Do the same in space. Describe the set of points in the plane given by 2 x 2 + y 2 4. Do the same in space. Philippe B. Laval (KSU) Rectangular Coordinates in Space Today 10 / 11
Exercises See the problems at the end of my notes on Rectangular Coordinates in Space. Philippe B. Laval (KSU) Rectangular Coordinates in Space Today 11 / 11