Math 314 Lecture #1 §12.1: Three-Dimensional Coordinate Systems Outcome A: Recall and apply the aspects of the three-dimensional rectangular coordi- nate system: coordinate axes, right-hand rule, coordinate planes, octants, first octant, coordinates of a point, and projections. To locate a point in three-dimensional space requires three real numbers arranged in an ordered triple (a, b, c). The origin O of three-dimensional space is the point (0, 0, 0). The coordinate axes are three directed lines through O that are perpendicular to each other: these are labeled the x-axis, the y-axis, and the z-axis. The right-hand rule determines the direction of the positive z-axes from the directions of the positive x- and y-axes: using your right hand, point your fingers in the positive x-axis direction, curl your fingers towards the positive y-axis, then your thumb points in the direction of the positive z-axes. Here are the three positive coordinate axes, according to the right-hand rule. The coordinate planes are determined by the coordinate axes: these are the xy-plane, the xz-plane, and the yz-plane. Here are the three coordinate planes. Which one is which? Eight octants are formed by the three coordinate planes. The first octant is determined by the positive axes. The coordinates of a point P are the ordered triple (a, b, c) where the value of a is the directed distance from the yz-plane to P , the value of b is the directed distance from the xz-plane to P , and the value of c is the directed distance from the xy-plane to P . The projection of the point (a, b, c) onto the xy-plane is the point (a, b, 0), its projection onto the xz-plane is the point (a, 0, c), and its projection onto the yz-plane is the point (0, b, c). The Cartesian product R3 = {(x, y, z): x, y, z ∈ R} is the set of all ordered triples of real numbers, which describes all points in 3-dimensional space by a rectangular coordi- nate system. Example. The point (1, 2, 3) in R3 is the point lies in the first octant, where starting from the origin, we move first 1 unit along the positive x-axis, then 2 units parallel to the y-axis, and finally 3 units parallel to the z-axis. Its coordinate plane projections are the points (1, 2, 0), (1, 0, 3), and (0, 2, 3), which along with the points (1, 0, 0), (0, 2, 0), and (0, 0, 3), the origin and (1, 2, 3), form the eight vertices of a rectangular box. Can you sketch this rectangular box? Outcome B: Describe and sketch (by hand) a surface in R3 represented by an equation in x, y, and z. Example. What surface in R3 does the equation z = 2 describe? It is the set {(x, y, 2) : y, z ∈ R} found by setting the z-coordinate of points to 2. This set is a horizontal plane. Example. What surface in R3 does the equation x + y = 1 describe? This is the set of points {(x, 1 − x, z): x, z ∈ R} found by solving x + y = 1 for y and setting the y-coordinate to this. This surface is a vertical plane that intersects the xy-plane along the line x + y = 1. Outcome C: Recall and apply the distance formula in three-dimensions. 3 The distance between the points P1 = (x1, y1, z1) and P2 = (x2, y2, z2) in R is p 2 2 2 |P1P2| = (x2 − x1) + (y2 − y1) + (z2 − z1) . Example. For the three points P = (2, −1, 0), Q = (4, 1, 1), and R = (4, −5, 4), we compute the distances √ |PQ| = p(4 − 2)2 + (1 − (−1))2 + (1 − 0)2 = 4 + 4 + 1 = 3, √ |PR| = p(4 − 2)2 + (−5 − (−1))2 + (4 − 0)2 = 4 + 16 + 16 = 6, √ √ |RQ| = p(4 − 4)2 + (−5 − 1)2 + (4 − 1)2 = 36 + 9 = 45. Is the triangle formed by the three points P , Q, and R an isosceles triangle? No. Does the triangle have a right angle? Yes, by the Pythagorean Theorem because |RQ|2 = |PQ|2 + |PR|2. Outcome D: Recall and apply the equation of a sphere. An equation of the sphere with center C = (h, k, l) and radius r is (x − h)2 + (y − k)2 + (z − l)2 = r2, i.e., the set of points a distance r from the point C. √ Example. The sphere (x−1)2+(y+1)2+z2 = 4 has center (1, −1, 0), radius r = 4 = 2, and intersects the xz-plane in the curve (x − 1)2 + (0 + 1)2 + z2 = 4, or (x − 1)2 + z2 = 3, which is obtained by setting y = 0. √ What is this curve in the xz-plane? A circle centered at (1, 0, 0) with radius 3. Outcome E: Correlate regions in R3 with inequalities in x, y, and z. The first octant in R3 is described by the inequalities x ≥ 0, y ≥ 0, and z ≥ 0. The inequalities 0 ≤ z ≤ 1 describes a horizontal slab contained between the horizontal planes z = 0 and z = 1..