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Instructor: Longfei Li Math 243 Lecture Notes 12.1 Three-Dimensional Coordinate Systems Much of what we will do is similar to Calculus I, but on higher dimensions. We will deal with functions with several variables instead of just a single variable as we did before. Previously, we mostly work in the 2D space, now we will work in 3D space. A point in a plane is represented as an ordered pair of real number (a, b); a is the x−coordinate, b is the y−coordinate In space, a point is represented as an ordered triple of real numbers (a, b, c); a is the x−coordinate, b is the y−coordinate and c is the z−coordinate. We need coordinate axes to represent points in space. x−, y−, z− axes follow the Right-Hand rule. Axes divide the space into 8 parts, called octants. Recall in a plane coordinate axes divide the plane into 4 parts, called quadrants. Figure 1: Coordinate Axes A point P in a space determines a rectangular box. From the box, we know the projection points of P onto the coordinate axes and coordinate planes (eg: xy−, yz− planes). Example1: The point (−4, 3, −5) determines a rectangular box as below. The projection onto the xz−plane is (−4, 0, −5). The projection onto the y−axis is (0, 3, 0). The distance to the y−axis is 1 √ 41. Notation: R3 = {(x, y, z)|x, y, z ∈ R} is the set of all triples of real numbers. 3D rectangular coordinate system: 1−to−1 Any point P in space ⇐⇒ ordered triples (a, b, c) ∈ R3 In 2D, the graph of an equation involving x, y is a curve in R2 . In 3D, the graph of an equation involving x, y, z is a surface in R3 Example 2: y = 5 is a plane in R3 while it represents a line in R2 . Remark: Context is important! Same equation represents different geometry in R2 and R3 Example 3: x2 + y 2 = 1 is a cylinder in R3 , while it’s a unit circle with center at the origin in R2 . Distance formula in 3D: The distance |AB| between the points A(x1 , y1 , z1 ) and B(x2 , y2 , z2 ) is p |AB| = (x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2 2 Equation of a sphere: (x − a)2 + (y − b2 ) + (z − c2 ) = r2 here P (a, b, c) is the center, r is the radius. 3