 12.1AAE: Submarine hunting Two surface ships on maneuvers are trying to dete...
 12.1PE: Vector Calculations in Two DimensionsIn Exercise, let Find (a) the ...
 12.2AAE: A helicopter rescue Two helicopters, H1 and H2 are traveling togeth...
 12.1QGY: When do directed line segments in the plane represent the same vector?
 12.2PE: Vector Calculations in Two DimensionsIn Exercise, let Find (a) the ...
 12.3AAE: Torque The operator’s manual for the Toro® 21 in. lawnmower says “t...
 12.2QGY: How are vectors added and subtracted geometrically? Algebraically?
 12.3PE: Vector Calculations in Two DimensionsIn Exercise, let Find (a) the ...
 12.3QGY: How do you find a vector’s magnitude and direction?
 12.4AAE: Rotating body The line through the origin and the point A(1, 1, 1) ...
 12.4PE: Vector Calculations in Two DimensionsIn Exercise, let Find (a) the ...
 12.4QGY: If a vector is multiplied by a positive scalar, how is the result r...
 12.5AAE: Consider the weight suspended by two wires in each diagram. Find th...
 12.5PE: Vector Calculations in Two DimensionsIn Exercise, find the componen...
 12.5QGY: Define the dot product (scalar product) of two vectors. Which algeb...
 12.6AAE: Consider a weight of w N suspended by two wires in the diagram, whe...
 12.6PE: Vector Calculations in Two DimensionsIn Exercise, find the componen...
 12.6QGY: What geometric interpretation does the dot product have? Give examp...
 12.7AAE: Determinants and planesa. Show that is an equation for the plane th...
 12.7PE: Vector Calculations in Two DimensionsIn Exercise, find the componen...
 12.7QGY: What is the vector projection of a vector u onto a vector v? Give a...
 12.8AAE: Determinants and lines Show that the lines1 1, 22, 33, And11, 22, 3...
 12.8PE: Vector Calculations in Two DimensionsIn Exercise, find the componen...
 12.8QGY: Define the cross product (vector product) of two vectors. Which alg...
 12.9AAE: Consider a regular tetrahedron of side length 2.a. Use vectors to f...
 12.9PE: Vector Calculations in Two DimensionsExpress the vectors in Exercis...
 12.9QGY: What geometric or physical interpretations do cross products have? ...
 12.10AAE: In the figure here, D is the midpoint of side AB of triangle ABC, a...
 12.10PE: Vector Calculations in Two DimensionsExpress the vectors in Exercis...
 12.10QGY: What is the determinant formula for calculating the cross product o...
 12.11AAE: Use vectors to show that the distance from P1(x1, y1) to the line a...
 12.11PE: Vector Calculations in Two DimensionsExpress the vectors in Exercis...
 12.11QGY: How do you find equations for lines, line segments, and planes in s...
 12.12AAE: a. Use vectors to show that the distance from P1(x1, y1, z1) to the...
 12.12PE: Vector Calculations in Two DimensionsExpress the vectors in Exercis...
 12.12QGY: How do you find the distance from a point to a line in space? From ...
 12.13AAE: a. Show that the distance between the parallel planes Ax + By + Cz ...
 12.13PE: Vector Calculations in Three DimensionsExpress the vectors in Exerc...
 12.13QGY: What are box products? What significance do they have? How are they...
 12.14AAE: Prove that four points A, B, C, and D are coplanar (lie in a common...
 12.14PE: Vector Calculations in Three DimensionsExpress the vectors in Exerc...
 12.14QGY: How do you find equations for spheres in space? Give examples.
 12.15AAE: The projection of a vector on a plane Let P be a plane in space and...
 12.15PE: Vector Calculations in Three DimensionsFind a vector 2 units long i...
 12.15QGY: How do you find the intersection of two lines in space? A line and ...
 12.16AAE: The accompanying figure shows nonzero vectors v, w, and z, with z o...
 12.16PE: Vector Calculations in Three DimensionsFind a vector 5 units long i...
 12.16QGY: What is a cylinder? Give examples of equations that define cylinder...
 12.17AAE: Triple vector products The triple vector products and are usually n...
 12.17PE: Vector Calculations in Three DimensionsIn Exercise, find the angle ...
 12.17QGY: What are quadric surfaces? Give examples of different kinds of elli...
 12.18AAE: Cross and dot products Show that if u, v, w, and r are any vectors,...
 12.18PE: Vector Calculations in Three DimensionsIn Exercise, find the angle ...
 12.19AAE: Cross and dot products Prove or disprove the formula
 12.19PE: Vector Calculations in Three DimensionsIn Exercise, find projv u.v ...
 12.20AAE: By forming the cross product of two appropriate vectors, derive the...
 12.20PE: Vector Calculations in Three DimensionsIn Exercise, find projv u.u ...
 12.21AAE: Use vectors to prove that for any four numbers a, b, c, and d. (Hin...
 12.21PE: Vector Calculations in Three DimensionsIn Exercise, draw coordinate...
 12.22AAE: Dot multiplication is positive definite Show that dot multiplicatio...
 12.22PE: Vector Calculations in Three DimensionsIn Exercise, draw coordinate...
 12.23AAE: Show that for any vectors u and v.
 12.23PE: Vector Calculations in Three DimensionsIf and the angle between v a...
 12.24PE: Vector Calculations in Three DimensionsFor what value or values of ...
 12.24AAE: Show that bisects the angle between u and v.
 12.25AAE: Show that are orthogonal.
 12.25PE: Vector Calculations in Three DimensionsIn Exercise, find (a) the ar...
 12.26PE: Vector Calculations in Three DimensionsIn Exercise, find (a) the ar...
 12.27PE: Lines, Planes, and DistancesSuppose that n is normal to a plane and...
 12.28PE: Lines, Planes, and DistancesFind a vector in the plane parallel to ...
 12.29PE: Lines, Planes, and DistancesIn Exercise, find the distance from the...
 12.30PE: Lines, Planes, and DistancesIn Exercise, find the distance from the...
 12.31PE: Lines, Planes, and DistancesParametrize the line that passes throug...
 12.32PE: Lines, Planes, and DistancesParametrize the line segment joining th...
 12.33PE: Lines, Planes, and DistancesIn Exercise, find the distance from the...
 12.34PE: Lines, Planes, and DistancesIn Exercise, find the distance from the...
 12.35PE: Lines, Planes, and DistancesFind an equation for the plane that pas...
 12.36PE: Lines, Planes, and DistancesFind an equation for the plane that pas...
 12.37PE: Lines, Planes, and DistancesIn Exercise, find an equation for the p...
 12.38PE: Lines, Planes, and DistancesIn Exercise, find an equation for the p...
 12.39PE: Lines, Planes, and DistancesFind the points in which the line x = 1...
 12.40PE: Lines, Planes, and DistancesFind the point in which the line throug...
 12.41PE: Lines, Planes, and DistancesFind the acute angle between the planes...
 12.42PE: Lines, Planes, and DistancesFind the acute angle between the planes...
 12.43PE: Lines, Planes, and DistancesFind parametric equations for the line ...
 12.45PE: Lines, Planes, and DistancesThe planes 3x + 6z = 1 and 2x + 2y  z ...
 12.44PE: Lines, Planes, and DistancesShow that the line in which the planesx...
 12.46PE: Lines, Planes, and DistancesFind an equation for the plane that pas...
 12.47PE: Lines, Planes, and DistancesIs v = 2i  4j + k related in any speci...
 12.48PE: Lines, Planes, and DistancesThe equation represent?
 12.49PE: Lines, Planes, and DistancesFind the distance from the point P(1, 4...
 12.50PE: Lines, Planes, and DistancesFind the distance from the point (2, 2,...
 12.51PE: Lines, Planes, and DistancesFind a vector parallel to the plane 2x ...
 12.52PE: Lines, Planes, and DistancesFind a unit vector orthogonal to A in t...
 12.53PE: Lines, Planes, and DistancesFind a vector of magnitude 2 parallel t...
 12.54PE: Lines, Planes, and DistancesFind the point in which the line throug...
 12.55PE: Lines, Planes, and DistancesFind the point in which the line throug...
 12.56PE: Lines, Planes, and DistancesWhat angle does the line of intersectio...
 12.57PE: Lines, Planes, and DistancesThe lineL: x = 3 + 2t, y = 2t, z = tint...
 12.58PE: Lines, Planes, and DistancesShow that for every real number k the p...
 12.59PE: Lines, Planes, and DistancesFind an equation for the plane through ...
 12.60PE: Lines, Planes, and DistancesIs the line x = 1 + 2t, y = 2 + 3t, z ...
 12.61PE: Lines, Planes, and DistancesWhich of the following are equations fo...
 12.62PE: Lines, Planes, and DistancesThe parallelogram shown here has vertic...
 12.63PE: Lines, Planes, and DistancesDistance between lines Find the distanc...
 12.64PE: Lines, Planes, and Distances(Continuation of Exercise 63.) Find the...
 12.65PE: Quadric SurfacesIdentify and sketch the surfaces in Exercise.
 12.66PE: Quadric SurfacesIdentify and sketch the surfaces in Exercise.
 12.67PE: Quadric SurfacesIdentify and sketch the surfaces in Exercise.
 12.68PE: Quadric SurfacesIdentify and sketch the surfaces in Exercise.
 12.69PE: Quadric SurfacesIdentify and sketch the surfaces in Exercise.
 12.70PE: Quadric SurfacesIdentify and sketch the surfaces in Exercise.
 12.71PE: Quadric SurfacesIdentify and sketch the surfaces in Exercise.
 12.72PE: Quadric SurfacesIdentify and sketch the surfaces in Exercise.
 12.73PE: Quadric SurfacesIdentify and sketch the surfaces in Exercise.
 12.74PE: Quadric SurfacesIdentify and sketch the surfaces in Exercise.
 12.75PE: Quadric SurfacesIdentify and sketch the surfaces in Exercise.
 12.76PE: Quadric SurfacesIdentify and sketch the surfaces in Exercise.
Solutions for Chapter 12: Thomas' Calculus: Early Transcendentals 13th Edition
Full solutions for Thomas' Calculus: Early Transcendentals  13th Edition
ISBN: 9780321884077
Solutions for Chapter 12
Get Full SolutionsChapter 12 includes 118 full stepbystep solutions. This textbook survival guide was created for the textbook: Thomas' Calculus: Early Transcendentals , edition: 13. This expansive textbook survival guide covers the following chapters and their solutions. Since 118 problems in chapter 12 have been answered, more than 81968 students have viewed full stepbystep solutions from this chapter. Thomas' Calculus: Early Transcendentals was written by and is associated to the ISBN: 9780321884077.

Acceleration due to gravity
g ? 32 ft/sec2 ? 9.8 m/sec

Boundary
The set of points on the “edge” of a region

Conditional probability
The probability of an event A given that an event B has already occurred

Equal complex numbers
Complex numbers whose real parts are equal and whose imaginary parts are equal.

Even function
A function whose graph is symmetric about the yaxis for all x in the domain of ƒ.

Factoring (a polynomial)
Writing a polynomial as a product of two or more polynomial factors.

Imaginary axis
See Complex plane.

Increasing on an interval
A function ƒ is increasing on an interval I if, for any two points in I, a positive change in x results in a positive change in.

Inferential statistics
Using the science of statistics to make inferences about the parameters in a population from a sample.

Initial side of an angle
See Angle.

Intermediate Value Theorem
If ƒ is a polynomial function and a < b , then ƒ assumes every value between ƒ(a) and ƒ(b).

Odd function
A function whose graph is symmetric about the origin (ƒ(x) = ƒ(x) for all x in the domain of f).

Quadratic regression
A procedure for fitting a quadratic function to a set of data.

Reexpression of data
A transformation of a data set.

Real zeros
Zeros of a function that are real numbers.

Rectangular coordinate system
See Cartesian coordinate system.

Seconddegree equation in two variables
Ax 2 + Bxy + Cy2 + Dx + Ey + F = 0, where A, B, and C are not all zero.

Solve a triangle
To find one or more unknown sides or angles of a triangle

Tangent line of ƒ at x = a
The line through (a, ƒ(a)) with slope ƒ'(a) provided ƒ'(a) exists.

Translation
See Horizontal translation, Vertical translation.