 11.1: In Exercises 1 and 2, let and and find (a) the component forms of u...
 11.2: In Exercises 1 and 2, let and and find (a) the component forms of u...
 11.3: In Exercises 3 and 4, find the component form of v given its magnit...
 11.4: In Exercises 3 and 4, find the component form of v given its magnit...
 11.5: Find the coordinates of the point in the plane four units to the ri...
 11.6: Find the coordinates of the point located on the axis and seven uni...
 11.7: In Exercises 7 and 8, determine the location of a point that satisf...
 11.8: In Exercises 7 and 8, determine the location of a point that satisf...
 11.9: In Exercises 9 and 10, find the standard equation of the sphere.
 11.10: In Exercises 9 and 10, find the standard equation of the sphere.
 11.11: In Exercises 11 and 12, complete the square to write the equation o...
 11.12: In Exercises 11 and 12, complete the square to write the equation o...
 11.13: In Exercises 13 and 14, the initial and terminal points of a vector...
 11.14: In Exercises 13 and 14, the initial and terminal points of a vector...
 11.15: In Exercises 15 and 16, use vectors to determine whether the points...
 11.16: In Exercises 15 and 16, use vectors to determine whether the points...
 11.17: Find a unit vector in the direction of
 11.18: Find the vector of magnitude 8 in the direction
 11.19: In Exercises 19 and 20, let and and find (a) the component forms of...
 11.20: In Exercises 19 and 20, let and and find (a) the component forms of...
 11.21: In Exercises 21 and 22, determine whether u and v are orthogonal, p...
 11.22: In Exercises 21 and 22, determine whether u and v are orthogonal, p...
 11.23: In Exercises 2326, find the angle between the vectors.
 11.24: In Exercises 2326, find the angle between the vectors.
 11.25: In Exercises 2326, find the angle between the vectors.
 11.26: In Exercises 2326, find the angle between the vectors.
 11.27: Find two vectors in opposite directions that are orthogonal to the ...
 11.28: Work An object is pulled 8 feet across a floor using a force of 75 ...
 11.29: In Exercises 2932, let and Show that
 11.30: In Exercises 2932, let and Find the angle between and
 11.31: In Exercises 2932, let and Determine the projection of onto
 11.32: In Exercises 2932, let and Find the work done in moving an object a...
 11.33: In Exercises 3338, let and Determine a unit vector perpendicular to...
 11.34: In Exercises 3338, let and Show that
 11.35: In Exercises 3338, let and Find the volume of the solid whose edges...
 11.36: In Exercises 3338, let and Show that
 11.37: In Exercises 3338, let and Find the area of the parallelogram with ...
 11.38: In Exercises 3338, let and Find the area of the triangle with adjac...
 11.39: Torque The specifications for a tractor state that the torque on a ...
 11.40: Volume Use the triple scalar product to find the volume of the para...
 11.41: In Exercises 41 and 42, find sets of (a) parametric equations and (...
 11.42: In Exercises 41 and 42, find sets of (a) parametric equations and (...
 11.43: In Exercises 4346, find (a) a set of parametric equations and (b) a...
 11.44: In Exercises 4346, find (a) a set of parametric equations and (b) a...
 11.45: In Exercises 4346, find (a) a set of parametric equations and (b) a...
 11.46: In Exercises 4346, find (a) a set of parametric equations and (b) a...
 11.47: In Exercises 4750, find an equation of the plane. The plane passes ...
 11.48: In Exercises 4750, find an equation of the plane. The plane passes ...
 11.49: In Exercises 4750, find an equation of the plane. The plane contain...
 11.50: In Exercises 4750, find an equation of the plane. The plane passes ...
 11.51: Find the distance between the point and the plane
 11.52: Find the distance between the point and the plane
 11.53: Find the distance between the planes and
 11.54: Find the distance between the point and the line given by and
 11.55: In Exercises 5564, describe and sketch the surface.
 11.56: In Exercises 5564, describe and sketch the surface.
 11.57: In Exercises 5564, describe and sketch the surface.
 11.58: In Exercises 5564, describe and sketch the surface.
 11.59: In Exercises 5564, describe and sketch the surface.
 11.60: In Exercises 5564, describe and sketch the surface.
 11.61: In Exercises 5564, describe and sketch the surface.
 11.62: In Exercises 5564, describe and sketch the surface.
 11.63: In Exercises 5564, describe and sketch the surface.
 11.64: In Exercises 5564, describe and sketch the surface.
 11.65: Find an equation of a generating curve of the surface of revolution
 11.66: Find an equation for the surface of revolution generated by revolvi...
 11.67: In Exercises 67 and 68, convert the point from rectangular coordina...
 11.68: In Exercises 67 and 68, convert the point from rectangular coordina...
 11.69: In Exercises 69 and 70, convert the point from cylindrical coordina...
 11.70: In Exercises 69 and 70, convert the point from cylindrical coordina...
 11.71: In Exercises 71 and 72, convert the point from spherical coordinate...
 11.72: In Exercises 71 and 72, convert the point from spherical coordinate...
 11.73: In Exercises 73 and 74, convert the rectangular equation to an equa...
 11.74: In Exercises 73 and 74, convert the rectangular equation to an equa...
 11.75: In Exercises 75 and 76, find an equation in rectangular coordinates...
 11.76: In Exercises 75 and 76, find an equation in rectangular coordinates...
 11.77: In Exercises 77 and 78, find an equation in rectangular coordinates...
 11.78: In Exercises 77 and 78, find an equation in rectangular coordinates...
 11.1: Using vectors, prove the Law of Sines: If and are the three sides o...
 11.2: Consider the function (a) Use a graphing utility to graph the funct...
 11.3: Using vectors, prove that the line segments joining the midpoints o...
 11.4: Using vectors, prove that the diagonals of a rhombus are perpendicu...
 11.5: (a) Find the shortest distance between the point and the line deter...
 11.6: Let be a point in the plane with normal vector Describe the set of ...
 11.7: (a) Find the volume of the solid bounded below by the paraboloid an...
 11.8: (a) Use the disk method to find the volume of the sphere (b) Find t...
 11.9: Sketch the graph of each equation given in spherical coordinates. (...
 11.10: Sketch the graph of each equation given in cylindrical coordinates....
 11.11: Prove the following property of the cross product.
 11.12: Consider the line given by the parametric equations and the point f...
 11.13: A tetherball weighing 1 pound is pulled outward from the pole by a ...
 11.14: A loaded barge is being towed by two tugboats, and the magnitude of...
 11.15: Consider the vectors and where Find the cross product of the vector...
 11.16: Los Angeles is located at North latitude and West longitude, and Ri...
 11.17: Consider the plane that passes through the points and Show that the...
 11.18: Show that the distance between the parallel planes and
 11.19: Show that the curve of intersection of the plane and the cylinder i...
 11.20: Read the article Tooth Tables: Solution of a Dental Vector Algebra ...
Solutions for Chapter 11: Vectors and the Geometry of Space
Full solutions for Calculus  8th Edition
ISBN: 9780618502981
Solutions for Chapter 11: Vectors and the Geometry of Space
Get Full SolutionsChapter 11: Vectors and the Geometry of Space includes 98 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Calculus was written by and is associated to the ISBN: 9780618502981. Since 98 problems in chapter 11: Vectors and the Geometry of Space have been answered, more than 79621 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Calculus, edition: 8.

Acute triangle
A triangle in which all angles measure less than 90°

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

Causation
A relationship between two variables in which the values of the response variable are directly affected by the values of the explanatory variable

Coefficient of determination
The number r2 or R2 that measures how well a regression curve fits the data

Common ratio
See Geometric sequence.

Control
The principle of experimental design that makes it possible to rule out other factors when making inferences about a particular explanatory variable

Differentiable at x = a
ƒ'(a) exists

Dihedral angle
An angle formed by two intersecting planes,

Direct variation
See Power function.

Imaginary unit
The complex number.

Inequality symbol or
<,>,<,>.

Inverse cosecant function
The function y = csc1 x

Mathematical induction
A process for proving that a statement is true for all natural numbers n by showing that it is true for n = 1 (the anchor) and that, if it is true for n = k, then it must be true for n = k + 1 (the inductive step)

Parametrization
A set of parametric equations for a curve.

Probability function
A function P that assigns a real number to each outcome O in a sample space satisfying: 0 … P1O2 … 1, P12 = 0, and the sum of the probabilities of all outcomes is 1.

Quartic regression
A procedure for fitting a quartic function to a set of data.

Reflection
Two points that are symmetric with respect to a lineor a point.

Sum of complex numbers
(a + bi) + (c + di) = (a + c) + (b + d)i

Time plot
A line graph in which time is measured on the horizontal axis.

zcoordinate
The directed distance from the xyplane to a point in space, or the third number in an ordered triple.