Chapter 18

August 20 2017

Chapter 18 Vectors in Three-Dimensional Space. Surfaces 18.1 Coordinates and Vectors in Three-Dimensional Space - so far we had focused on the calculus of functions of a single variable - the geometry of these functions is two-dimensional because the graph of a function of a single variable is a curve in the plane - for the remainder of the book we focus on the calculus of functions of several (two or more) independent variables - the geometry of functions of two variables is three-dimensional - because in general the graph of such a function is a curved surface in space - in this chapter we discuss the analytic geometry of three-dimensional space - our treatment will emphasize vector algebra - partly because this approach provides a more direct and intuitive understanding of the equations of lines and planes - partly because the concepts of dots and cross products are indispensable in many other parts of mathematics and physics - rectangular coordinates in the plane can be generalized in a natural way to rectangular coordinates in space - just as in plane analytic geometry - we write P = (x, y, z) and identify the point P with the ordered triple of its coordinates - the xy-plane is the set of all points - (x, y, 0), or z = 0 - the z-axis is the set of all points - (0, 0, z), represented by the pair of equations x = 0 and y = 0 - they are the equations of the yz-plane and the xz-plane - taken together they characterize the z-axis as the intersection of these two coordinate planes - almost all the ideas about vectors that were presented before are valid in three-dimensional space - if P = (x, y, z) is any point in space - the position vector - R = OP can be written - R = xi + yj + zk - the distance between any two points is - P1P2 = sqrt (x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2 - the equation of a sphere can be written as - | P0P | = r - (x - x0)2 + (y - y0)2 + (z - z0)2 = r2 18.2 The Dot Product of Two Vectors - there are two different ways of defining the product of two vectors - the dot product A dot B is a scalar number - the cross product A x B is a vector - A dot B = |A| |B| cos angle - |B| cos angle is the scalar projection of B on A, proj A B - A dot B = length of A x scalar projection of B on A = length of B x scalar projection of A on B - i dot i = j dot j = k dot k = 1 - i dot j = j dot k = i dot k = 0 - A dot B = a1b1 + a2b2 + a3b3 - cos angle = A dot B / |A| |B| - the main significance of the dot product in geometry - is it provides a simple way to find the angle between two vectors - and to determine when two vectors are perpendicular - A per B if and only if A dot B = 0 - the simplest physical illustration of the use of the dot product is the concept of work - the work done by a force exerted along the line of motion 18.3 The Cross Product of Two Vectors - many problems in geometry require us to find a vector perpendicular to each of two given vectors - a routine way of doing this is provided by the cross product A x B - vectors A and B determine the plane under consideration - they also determine a parallelogram in this plane - of area |A| |B| sin angle - we take this area to be the magnitude of the vector A x B - A x B = |A| |B| sin angle n - n is the unit vector whose direction is determined by the right-hand rule - A is parallel to B only if A x B = 0 - B x A = - A x B - cross product is not commutative - i x j = - j x i = k - j x k = - k x j = i - k x i = - i x k = j - i x i = j x j = k x k = 0 - the cross product is distributive, so - A x B = i (a2b3 - a3b2) - j (a1b3 - a3b1) + k (a1b2 - a2b1) - the formula is easier to remember with determinants 18.4 Lines and Planes - since all the machinery of vector algebra is now in place - we now turn to the calculus of vector functions in three-dimensional space - R(t) = x(t) i + y(t) j + z(t) k - dR/dt is tangent to the path at point P - is the velocity of P if t is time - is the unit tangent vector if t is arc length - we put aside the calculus of vector functions to focus on the analytic geometry of lines, planes, and curved surfaces in three-dimensional space - in plane analytic geometry a single first-degree equation - ax + by + c = 0 - is the equation of a straight line - in three dimensions such an equation represents a plane - it is not possible to represent a line by any single first-degree equation - a line in space can be given geometrically in three ways - a line through two points - the intersection of two planes - the line through a point in a specified direction - the third way is the most important for us - suppose L is the line in space that passes through a given point P0 = (x0, y0, z0) and is parallel to a given nonzero vector - V = ai + bj + ck - then another point P = (x, y, z) lies on L - if and only if P0P is parallel to V - that is - P lies on L if and only if P0P is a scalar multiple of V - P0P = tV - if R0 = OP0 and R = OP are the position vectors of P0 and P - then P0P = R - R0 - R = R0 + tV - which is the vector equation of L - as t varies from -inf to inf - the point P traverses the entire line L - moving in the direction of V - if we rewrite as - xi + yj + zk = x0i + y0j + z0k + t(ai + bj + ck) - then we get the three scalar equations - x = x0 + at - y = y0 + bt - z = z0 + ct - these are the parametric equations of the line L through point P0 = (x0, y0, z0) - and parallel to vector V = ai + bj + ck - in order to obtain the Cartesian equations of the line - we eliminate the parameters by solving for t - x - x0 / a = y - y0 / b = z - z0 / c - these are the symmetric equations of the line L - we now turn to the study of planes - a plane can also be characterized in three ways - as the plane through three noncollinear points - as the plane through a line and a point not on the line - as the plane through a point and perpendicular to a specified direction - again, the third approach is the most convenient for us - consider the plane that passes through P0 = (x0, y0, z0) - and is perpendicular to nonzero vector - N = ai + bj + ck - another point P = (x, y, z) lies on this plane - if and only if P0P is perpendicular to N - N dot P0P = 0 - N dot (R - R0) = 0 - this is the vector equation of the plane - a (x - x0) + b (y - y0) + c (z - z0) = 0 - this is the Cartesian equation of the plane through P0 with normal vector N - ax + by + cz = d - d = ax0 + by0 + cz0 - the coefficients a, b, c of x, y, z are the components of normal vector N - every linear equation in x, y, z represents a plane with normal vector N = ai + bj + ck 18.5 Cylinders and Surfaces of Revolution - we know the graph of an equation f(x, y) = 0 is usually a curve in the xy-plane - in the same way - F (x, y, z) = 0 is usually a surface in xyz-space - the simplest surfaces are planes - containing only first-degree terms in the variables - ax + by + cz + d = 0 - cylinders are the next surfaces after planes in order of complexity - consider a plane curve C and line L not parallel to the plane of C - a cylinder is the geometric figure in space generated by a straight line moving parallel to L and passing through C - the moving line is a generator - the cylinder can be thought of as consisting of infinitely many parallel lines, called rulings - if the generator is the z-axis - the same equation for a curve f(x, y) = 0 is the equation for the three-dimensional cylinder - the two-dimensional ellipse can be extended in three-dimensions as an elliptic cylinder - any equation in rectangular coordinates x, y, z with one variable missing - represents a cylinder whose rulings are parallel to the axis corresponding to the missing variable - another way to generate a surface by using a plane curve C - is to revolve the curve about a line L in its plane - the revolving surface is called a surface of revolution with axis L 18.6 Quadric Surfaces - in section 15.6 we learned the graph of a second-degree equation in the variables x and y - is always a conic section - a parabola, an ellipse, or a hyperbola - in three-dimensional space the most general equation of the second degree - is a quadric surface - there are exactly six distinct kinds of nondegenerate quadric surfaces - the ellipsoid - the hyperboloid of one sheet - the hyperboloid of two sheets - the elliptic cone - the elliptic paraboloid - the hyperbolic paraboloid 18.7 Cylindrical and Spherical Coordinates - in plane analytic geometry we used a rectangular coordinate system for some types of problems - and a polar coordinate system fo rothers - we saw there are many situations in which one system is more convenient than the other - the same is true for the study of geometry and calculus in space - we now describe two other three-dimensional coordinate systems - in addition to the now familiar rectangular coordinate system - the cylindrical coordinates of a point P whose rectangular coordinates are (x, y, z) - is obtained by replacing x and y with the corresponding polar coordinates of r and angle - allowing z to remain unchanged - the term cylindrical is used because the graph of r = a is a right circular cylinder - in physics - cylindrical coordinates are particularly convenient for studying situations in which there is axial symmetry - the spherical coordinates of P are (p, angle, angle) - the term spherical is used because the graph of p = a is a sphere - there are many physical uses of spherical coordinates - ranging from problems about heat conduction to problems in the theory of gravitation