12.2: Three-Dimensional Space Surfaces, Vectors, and Curves

University:
The Pennsylvania State University
Course:
Math 231 | Calculus Severable Variables
Academic year:

2024
Views:

88

Pages:

2
Author:

Alexis H.

12.2 Three-Dimensional Space: Surfaces, Vectors, and Curves By convention, we label the axes x, y, and z. This labeling satisfies the right-hand rule, which means that when you position your right hand so that your fingers curl from the positive x-axis toward the positive y-axis, your thumb points in the positive edirection We denote the set of all triples (a, b, c) by R^3, and we refer to this set as 3-space or three dimensional space. The coordinate planes in R^3 are defined by setting one of the coordinates equal to zero The xy-plane consists of the points (a, b, 0) and is defined by the equation z = 0. Similarly, x = 0 defines the yz-plane consisting of the points (0, b, c), and y = 0 defines the xz-plane consisting of the points (a, 0, c) z y Theorem 1: Distance formula in R^3 The distance |P-Q| between the points P = (a1, b1, c1) and Q = (a2, b2, c2) is jP ¡ Qj = q 2 2 2 (a2 ¡ a1) + (b2 ¡ b1) + (c2 ¡ c1) Surfaces The equation x = a defines a plane parallel to the yz-plane, while y = b and z = c defines the planes parallel to the xz-plane and yz-plane, respectively Spheres and cylinders are some other surfaces we will encounter. The sphere of radius R with center Q = (a, b, c) consists of all points P = (x, y, z) located a distance R from Q q 2 2 2 (x ¡ a) + (y ¡ b) + (z ¡ c) = R In R^2, a circle = 2 In R^3, a right circular cylinder = In R^3, a sphere = 2 f(x; y) : (x ¡ a) + (y ¡ b) = R2 g 2 2 2 f(x; y; z) : (x ¡ a) + (y ¡ b) = R2 g 2 2 (x ¡ a) + (y ¡ b) + (z ¡ c) = R2 Vectors in 3-Space If P = (a1, b1, c1) and Q = (a2, b2, c2), then the length or magnitude of v = PQ is the distance from P to Q: °¡¡!° q 2 2 2 ° ° kvk = °P Q° = (a2 ¡ a1) + (b2 ¡ b1) + (c2 ¡ c1) A vector is said to undergo a translation if It is moved without changing direction or magnitude Two vectors v and w are equivalent if w is a translation of v; that is, v and w have the same length and direction The position vector of a point Q(0) is the vector v(0) = OQ(0) based at the origin A vector v = PQ with components is equivalent to the vector v(0) = OQ(0) based at the origin with Q(0) = (a, b, c) The components of v = PQ, where P = (a1, b1, c1) and Q = (a2, b2, c2), are the differences a = a2-a1, b = b2 - b1, c = c2 - c1; that is, v = P Q = ha; b; ci = (a2 ¡ a1; b2 ¡ b1; c2 ¡ c1) Two vectors are equivalent if and only if they have the same components Vector addition and scalar multiplication are defined as in the the two-dimensional case. Vector addition is defined by the parallelogram law Vector addition is communicative, is associative, and satisfies the distributive property with respect to scalar multiplication The standard basis vectors in R^3 are: i = <1, 0, 0>. j = <0, 1, 0>. k = <0, 0, 1> Curves and Lines Alternatively, taken together, the equations for x (t), y (t), z (t) form the components of a vector r (t) = with base at the origin Since there are many lines in any given direction (all parallel) we need to specify a point on the line to uniquely determine the line. Thus, to represent a line in 3-space we use a direction vector and a point on the line Given A direction vector v = for a line, and a point P(0) = (x (0), y (0), z (0)) on it, we first describe the line geometrically, and then translate that to two algebraic representations, one as a vector equation and another as z parametric equations in x, y, z (x0; y0; z0) ha; b; ci Vector parametrization: r (t) = r0 + tv = hx0; y0; z0i + tha; b; ci Parametric equations: Y x (t) = x0 + at y (t) = y0 + bt z (t) = z0 + ct Two lines in R^3 coincide if they are parallel and pass through a common point, so we can always check whether two parameterizations describe the same line