 12.1: What is the difference between a vector and a scalar?
 12.2: How do you add two vectors geometrically? How do you add them algeb...
 12.3: If a is a vector and c is a scalar, how is ca related to a geometri...
 12.4: How do you find the vector from one point to another?
 12.5: How do you find the dot product of two vectors if you know their le...
 12.6: How are dot products useful?
 12.7: Write expressions for the scalar and vector projections of b onto a...
 12.8: How do you find the cross product a b of two vectors if you know th...
 12.9: How are cross products useful?
 12.10: (a) How do you find the area of the parallelogram determined by a a...
 12.11: How do you find a vector perpendicular to a plane?
 12.12: How do you find the angle between two intersecting planes?
 12.13: Write a vector equation, parametric equations, and symmetric equati...
 12.14: Write a vector equation and a scalar equation for a plane.
 12.15: (a) How do you tell if two vectors are parallel? (b) How do you tel...
 12.16: a) Describe a method for determining whether three points , , and l...
 12.17: (a) How do you find the distance from a point to a line? (b) How do...
 12.18: What are the traces of a surface? How do you find them?
 12.19: Write equations in standard form of the six types of quadric surfaces.
 12.20: (a) Write the equations for converting from cylindrical to rectangu...
 12.21: Find the point in which the line with parametric equations , , inte...
 12.22: Find the distance from the origin to the line x 1 t2 y 2 t z 1 2tx 1
 12.23: Determine whether the lines given by the symmetric equations and ar...
 12.24: (a) Show that the planes and are neither parallel nor perpendicular...
 12.25: Find the distance between the planes Find the distance between the ...
 12.26: Identify and sketch the graph of each surface.
 12.27: Identify and sketch the graph of each surface.
 12.28: Identify and sketch the graph of each surface.
 12.29: Identify and sketch the graph of each surface.
 12.30: Identify and sketch the graph of each surface.
 12.31: Identify and sketch the graph of each surface.
 12.32: Identify and sketch the graph of each surface.
 12.33: Identify and sketch the graph of each surface.
 12.34: Identify and sketch the graph of each surface.
 12.35: An ellipsoid is created by rotating the ellipse about the axis. Fi...
 12.36: A surface consists of all points such that the distance from to the...
 12.37: The cylindrical coordinates of a point are . Find the rectangular a...
 12.38: The rectangular coordinates of a point are . Find the cylindrical a...
 12.39: The spherical coordinates of a point are . Find the rectangular and...
 12.40: Identify the surface whose equation is given.
 12.41: Identify the surface whose equation is given.
 12.42: Identify the surface whose equation is given.
 12.43: Identify the surface whose equation is given.
 12.44: Write the equation in cylindrical coordinates and in spherical coor...
 12.45: Write the equation in cylindrical coordinates and in spherical coor...
 12.46: Write the equation in cylindrical coordinates and in spherical coor...
 12.47: The parabola , is rotated about the axis. Write an equation of the...
 12.48: Sketch the solid consisting of all points with spherical coordinate...
Solutions for Chapter 12: Vectors and the Geometry of Space
Full solutions for Calculus,  5th Edition
ISBN: 9780534393397
Solutions for Chapter 12: Vectors and the Geometry of Space
Get Full SolutionsChapter 12: Vectors and the Geometry of Space includes 48 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Calculus,, edition: 5. Since 48 problems in chapter 12: Vectors and the Geometry of Space have been answered, more than 44740 students have viewed full stepbystep solutions from this chapter. Calculus, was written by and is associated to the ISBN: 9780534393397.

Absolute value of a vector
See Magnitude of a vector.

Acute angle
An angle whose measure is between 0° and 90°

Blocking
A feature of some experimental designs that controls for potential differences between subject groups by applying treatments randomly within homogeneous blocks of subjects

Conic section (or conic)
A curve obtained by intersecting a doublenapped right circular cone with a plane

Coterminal angles
Two angles having the same initial side and the same terminal side

Difference of functions
(ƒ  g)(x) = ƒ(x)  g(x)

Distance (on a number line)
The distance between real numbers a and b, or a  b

Fibonacci numbers
The terms of the Fibonacci sequence.

Head minus tail (HMT) rule
An arrow with initial point (x1, y1 ) and terminal point (x2, y2) represents the vector <8x 2  x 1, y2  y19>

Magnitude of a real number
See Absolute value of a real number

Nonsingular matrix
A square matrix with nonzero determinant

Order of magnitude (of n)
log n.

Partial fraction decomposition
See Partial fractions.

Reexpression of data
A transformation of a data set.

Reference angle
See Reference triangle

Supply curve
p = ƒ(x), where x represents production and p represents price

Trigonometric form of a complex number
r(cos ? + i sin ?)

Upper bound test for real zeros
A test for finding an upper bound for the real zeros of a polynomial.

Variance
The square of the standard deviation.

Work
The product of a force applied to an object over a given distance W = ƒFƒ ƒAB!ƒ.