**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