 12.1: In Exercises 14, let and Find (a) the component form of the vector ...
 12.2: In Exercises 14, let and Find (a) the component form of the vector ...
 12.3: In Exercises 14, let and Find (a) the component form of the vector ...
 12.4: In Exercises 14, let and Find (a) the component form of the vector ...
 12.5: In Exercises 58, find the component form of the vector
 12.6: In Exercises 58, find the component form of the vector
 12.7: In Exercises 58, find the component form of the vector
 12.8: In Exercises 58, find the component form of the vector
 12.9: Express the vectors in Exercises 912 in terms of their lengths and ...
 12.10: Express the vectors in Exercises 912 in terms of their lengths and ...
 12.11: Express the vectors in Exercises 912 in terms of their lengths and ...
 12.12: Express the vectors in Exercises 912 in terms of their lengths and ...
 12.13: Express the vectors in Exercises 13 and 14 in terms of their length...
 12.14: Express the vectors in Exercises 13 and 14 in terms of their length...
 12.15: Find a vector 2 units long in the direction of
 12.16: Find a vector 5 units long in the direction opposite to the direction
 12.17: In Exercises 17 and 18, find the angle between v and u, the scalar ...
 12.18: In Exercises 17 and 18, find the angle between v and u, the scalar ...
 12.19: In Exercises 19 and 20, find projv u
 12.20: In Exercises 19 and 20, find projv u
 12.21: In Exercises 21 and 22, draw coordinate axes and then sketch u, v, ...
 12.22: In Exercises 21 and 22, draw coordinate axes and then sketch u, v, ...
 12.23: If and the angle between v and w is find
 12.24: For what value or values of a will the vectors and be v = 4i  8j ...
 12.25: In Exercises 25 and 26, find (a) the area of the parallelogram dete...
 12.26: In Exercises 25 and 26, find (a) the area of the parallelogram dete...
 12.27: Suppose that n is normal to a plane and that v is parallel to the p...
 12.28: Find a vector in the plane parallel to the line
 12.29: In Exercises 29 and 30, find the distance from the point to the line.
 12.30: In Exercises 29 and 30, find the distance from the point to the line.
 12.31: Parametrize the line that passes through the point (1, 2, 3) parall...
 12.32: Parametrize the line segment joining the points P(1, 2, 0) and
 12.33: In Exercises 33 and 34, find the distance from the point to the plane.
 12.34: In Exercises 33 and 34, find the distance from the point to the plane.
 12.35: Find an equation for the plane that passes through the point normal...
 12.36: Find an equation for the plane that passes through the point perpen...
 12.37: In Exercises 37 and 38, find an equation for the plane through poin...
 12.38: In Exercises 37 and 38, find an equation for the plane through poin...
 12.39: Find the points in which the line meets the three coordinate planes
 12.40: Find the point in which the line through the origin perpendicular t...
 12.41: Find the acute angle between the planes and
 12.42: Find the acute angle between the planes and
 12.43: Find parametric equations for the line in which the planes and inte...
 12.44: Show that the line in which the planes intersect is parallel to the...
 12.45: The planes and intersect in a line. a. Show that the planes are ort...
 12.46: Find an equation for the plane that passes through the point (1, 2,...
 12.47: Is related in any special way to the plane Give reasons for your an...
 12.48: The equation represents the plane through normal to n. What set doe...
 12.49: Find the distance from the point P(1, 4, 0) to the plane through A(...
 12.50: Find the distance from the point (2, 2, 3) to the plane
 12.51: Find a vector parallel to the plane and orthogonal to
 12.52: Find a unit vector orthogonal to A in the plane of B and C if
 12.53: Find a vector of magnitude 2 parallel to the line of intersection o...
 12.54: Find the point in which the line through the origin perpendicular t...
 12.55: Find the point in which the line through P(3, 2, 1) normal to the p...
 12.56: What angle does the line of intersection of the planes and make wit...
 12.57: The line intersects the plane in a point P. Find the coordinates of...
 12.58: Show that for every real number k the plane contains the line of in...
 12.59: Find an equation for the plane through and that lies parallel to th...
 12.60: Is the line related in any way to the plane Give reasons for your a...
 12.61: Which of the following are equations for the plane through the poin...
 12.62: The parallelogram shown here has vertices at and D. Find a. the coo...
 12.63: Find the distance between the line through the points and and the l...
 12.64: (Continuation of Exercise 63.) Find the distance between the line t...
 12.65: Identify and sketch the surfaces in Exercises 6576.
 12.66: Identify and sketch the surfaces in Exercises 6576.
 12.67: Identify and sketch the surfaces in Exercises 6576.
 12.68: Identify and sketch the surfaces in Exercises 6576.
 12.69: Identify and sketch the surfaces in Exercises 6576.
 12.70: Identify and sketch the surfaces in Exercises 6576.
 12.71: Identify and sketch the surfaces in Exercises 6576.
 12.72: Identify and sketch the surfaces in Exercises 6576.
 12.73: Identify and sketch the surfaces in Exercises 6576.
 12.74: Identify and sketch the surfaces in Exercises 6576.
 12.75: Identify and sketch the surfaces in Exercises 6576.
 12.76: Identify and sketch the surfaces in Exercises 6576.
Solutions for Chapter 12: Vectors and the Geometry of Space
Full solutions for Thomas' Calculus Early Transcendentals  12th Edition
ISBN: 9780321588760
Solutions for Chapter 12: Vectors and the Geometry of Space
Get Full SolutionsThomas' Calculus Early Transcendentals was written by and is associated to the ISBN: 9780321588760. This textbook survival guide was created for the textbook: Thomas' Calculus Early Transcendentals, edition: 12. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 12: Vectors and the Geometry of Space includes 76 full stepbystep solutions. Since 76 problems in chapter 12: Vectors and the Geometry of Space have been answered, more than 32979 students have viewed full stepbystep solutions from this chapter.

Continuous at x = a
lim x:a x a ƒ(x) = ƒ(a)

Distance (in Cartesian space)
The distance d(P, Q) between and P(x, y, z) and Q(x, y, z) or d(P, Q) ((x )  x 2)2 + (y1  y2)2 + (z 1  z 2)2

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

Frequency table (in statistics)
A table showing frequencies.

Future value of an annuity
The net amount of money returned from an annuity.

Hypotenuse
Side opposite the right angle in a right triangle.

Integrable over [a, b] Lba
ƒ1x2 dx exists.

Linear correlation
A scatter plot with points clustered along a line. Correlation is positive if the slope is positive and negative if the slope is negative

Linear regression
A procedure for finding the straight line that is the best fit for the data

NDER ƒ(a)
See Numerical derivative of ƒ at x = a.

Polar coordinates
The numbers (r, ?) that determine a point’s location in a polar coordinate system. The number r is the directed distance and ? is the directed angle

Positive association
A relationship between two variables in which higher values of one variable are generally associated with higher values of the other variable, p. 717.

Probability distribution
The collection of probabilities of outcomes in a sample space assigned by a probability function.

Quotient rule of logarithms
logb a R S b = logb R  logb S, R > 0, S > 0

Replication
The principle of experimental design that minimizes the effects of chance variation by repeating the experiment multiple times.

Terminal point
See Arrow.

Terms of a sequence
The range elements of a sequence.

Third quartile
See Quartile.

Trichotomy property
For real numbers a and b, exactly one of the following is true: a < b, a = b , or a > b.

Vertical line
x = a.