12.2 Three-dimensional space: Surfaces, vectors, and curves By convention, we label the axes where the positive sides of are labeled x, y, and z. This labeling satisfies the righthand 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 e-direction Each point in space has unique coordinates (a, b, c) relative to the axes. 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 divide r^3 into eight octanes (analogous to the four quadrants in the plane). Each octant corresponds to a possible combination of signs of the coordinates Surfaces theorem 1: Planes are the most basic surfaces. The equation x=a defines a plane parallel to the yz-plane, while y=b and z=c defines planes parallel to the xz-plane and The distance |P - q| between the xy-plane, respectively points P = (a1, b1, c1) and q = (a2, On squaring both sides, we obtain the standard equation of the sphere: b2, c2) is jP ¡ Qj = q 2 2 2 (a2 ¡ a; ) + (b2 ¡ b1) + (c2 ¡ c 1)2 2 2 (x ¡ a) + (y ¡ b) + (z ¡ c) = R2 Equation of a cylinder: 2 2 (x ¡ a) + (y ¡ b) = R2 Vectors in 3-space As in the plane, a vector v = pq in R^3 is determined by an initial point P and a terminal point Q The terminology and basic properties discussed in the previous section carry over to r^3 with little change. A vector V 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 with components is equivalent to the vector v based at the origin with q(0) = (a, b, c) Two vectors are equivalent if and only if they have the same components Vector addition and scalar multiplication are defined as in the two-dimensional case. Vector addition is defined by the parallelogram law Vector addition is commutative, is associative, and satisfies the distributive property with respect to scalar multiplication Curves and lines Most curves in 3-space that we consider will be expressed parametrically. That is, we will describe a curve using a set of three equations for x(t), y(t), z(t) that we can think of as representing the coordinates of a particle traveling through space and tracing out a curve In the plane, a line is identified by a slope (direction) and a point on the line. The same idea carries over to lines in r^3, but instead we use a direction vector parallel to the line to determine direction Since there are many lines in any given direction (all parallel) we need to specify a point on the line uniquely determine the line. Thus, to represent a line in 3-space we use a direction vector and a point on the line Vector parametrization r(t): r (t) = r0 + tv = hx0; y0; z0i + tha; b; ci Parametric equation: x (t) = x0 + at; y (t) = y0 + bt; z (t) = z0 + ct