 12.12.1: In Exercises 14, let u = (3,4) and v = (2, 5). Find (a) the comp...
 12.12.2: In Exercises 14, let u = (3,4) and v = (2, 5). Find (a) the comp...
 12.12.3: In Exercises 14, let u = (3,4) and v = (2, 5). Find (a) the comp...
 12.12.4: In Exercises 14, let u = (3,4) and v = (2, 5). Find (a) the comp...
 12.12.5: In Exercises 58, fmd the component form of the vector The veetor o...
 12.12.6: In Exercises 58, fmd the component form of the vector The unit vec...
 12.12.7: In Exercises 58, fmd the component form of the vector The vector 2...
 12.12.8: In Exercises 58, fmd the component form of the vector The veetor 5...
 12.12.9: Express the vectors in Exercises 912 in terms of their lengtha aod...
 12.12.10: Express the vectors in Exercises 912 in terms of their lengtha aod...
 12.12.11: Express the vectors in Exercises 912 in terms of their lengtha aod...
 12.12.12: Express the vectors in Exercises 912 in terms of their lengtha aod...
 12.12.13: Express the veetors in Exercises 13 and 14 in terms of their leogth...
 12.12.14: Express the veetors in Exercises 13 and 14 in terms of their leogth...
 12.12.15: Find a vector 2 units long in the direction ofv = 4i  j + 4k.
 12.12.16: Find a vector 5 units loog in the direction opposite to the directi...
 12.12.17: In Esercises 17 and 18, fmd lvi, lui, V'u, U'v, v X u, u X v, I v X...
 12.12.18: In Esercises 17 and 18, fmd lvi, lui, V'u, U'v, v X u, u X v, I v X...
 12.12.19: In Exercises 19 and 20, fmd proj. u. v=2i+jk
 12.12.20: In Exercises 19 and 20, fmd proj. u. u=i2j
 12.12.21: In Exercises 21 and 22, draw coordinate axes and then sketch u, v, ...
 12.12.22: In Exercises 21 and 22, draw coordinate axes and then sketch u, v, ...
 12.12.23: !flvl = 2, Iwl = 3,andthe angie between v andw is 1f/3,fmd Iv  2wl
 12.12.24: For what value or values of a will the vectors u = 2i + 4j  5k and...
 12.12.25: In Exercises 25 and 26, fmd (a) the area of the parallelogram deter...
 12.12.26: In Exercises 25 and 26, fmd (a) the area of the parallelogram deter...
 12.12.27: Suppose that n is DOnnai to a plane and that v is parallel to the p...
 12.12.28: Find a veetor in the plane parallel to the line ax + by = c.
 12.12.29: In Exercises 29 and 30, fmd the distance from the point to the line...
 12.12.30: In Exercises 29 and 30, fmd the distance from the point to the line...
 12.12.31: Parametrize the line that passes througb the point (I, 2, 3) parall...
 12.12.32: Parametrize the line segment joining the points P(I, 2, 0) and Q(I,...
 12.12.33: In Esercises 33 and 34, fmd the distance from the point to the plan...
 12.12.34: In Esercises 33 and 34, fmd the distance from the point to the plan...
 12.12.35: Find an equation for the plane that passes through the point (3, 2...
 12.12.36: Find an equatioo for the plane that passes through the point (1,6,...
 12.12.37: In Exercises 37 and 38, fmd an equatioo for the plane througb point...
 12.12.38: In Exercises 37 and 38, fmd an equatioo for the plane througb point...
 12.12.39: Find the points in which the line x = I + 21, y = I  I, z = 3t me...
 12.12.40: Find the point in which the line througb the origin perpendicular t...
 12.12.41: Find the acute angle between the planes x = 7 and x + y + v2z = 3.
 12.12.42: Find the acute angle betweeo the planes x + y = I and y+z= 1.
 12.12.43: Find pararne1ric equations for the line in which the planes x + 2y ...
 12.12.44: Show that the line in which the planes x+2y2z=5 and 5x2yz=0 inte...
 12.12.45: The planes 3x + 6z = I and 2x + 2y  z = 3 intcncct in a Iinc. L Sh...
 12.12.46: Find an equation for the plane that pasSCll through the point (1,2,...
 12.12.47: Is v = 2i  4j + k related in any !pCcial way to the plane 2x + Y =...
 12.12.48: The equation.' P;P = 0 represents ~plane through Po normal to D. W...
 12.12.49: Find the distance from the point P(I, 4, 0) to the plane through A(...
 12.12.50: Find the distance from the point (2, 2, 3) to the plane 2x + 3y + 5...
 12.12.51: Find a vector parallel to the plane 2x  y  z  4 and orthogonalto...
 12.12.52: Find a unit vector orthogonal to A in the plane of B and C if A  2...
 12.12.53: Find a vector of magnitude 2 parallel to the line of intcncction of...
 12.12.54: Find the point in which the line through the origin perpendicular t...
 12.12.55: Find the point in which the line through p(3, 2, I) normal ttl the ...
 12.12.56: What angle doc! the line of intcncction of the planca 2x + Y  z = ...
 12.12.57: The line L: x=3+2t, y=2t, z=t in1eDcctll the plane x + 3y  z = 4 ...
 12.12.58: Shaw that for every real number k the plane x  2y+ z + 3 + k(2x  ...
 12.12.59: Find an equation for the plane through A(2,O, 3) and B(I, 2, I) ...
 12.12.60: Is the line x = 1 + 2t, y = 2 + 31, z = 5t related in any way to ...
 12.12.61: Which of the following are equations for the plane through the poin...
 12.12.62: The puallelogram !hawn here has vertices at A(2, 1,4), B(I, 0, I)...
 12.12.63: Find the distance bctwccn the line L, through the points .4(1,0,1)...
 12.12.64: (Continuatio1l o/Exerci.!e 63.) Find the distance between the line ...
 12.12.65: Identify and sketch the surfaces in Exercises 6576. x2 + y2 + z2 = 4
 12.12.66: Identify and sketch the surfaces in Exercises 6576. x2 + (y  1)2 ...
 12.12.67: Identify and sketch the surfaces in Exercises 6576. 4x2+4y 2+ Z l=4
 12.12.68: Identify and sketch the surfaces in Exercises 6576. 3fu:l + 912 + ...
 12.12.69: Identify and sketch the surfaces in Exercises 6576. z = _(x2 + y2)
 12.12.70: Identify and sketch the surfaces in Exercises 6576. Y = _(Xl + zZ)
 12.12.71: Identify and sketch the surfaces in Exercises 6576. X2 +y2=z2
 12.12.72: Identify and sketch the surfaces in Exercises 6576. Xl+zZ=yl
 12.12.73: Identify and sketch the surfaces in Exercises 6576. X2 +y2_z2=4
 12.12.74: Identify and sketch the surfaces in Exercises 6576. 4yl+zZob:l =4
 12.12.75: Identify and sketch the surfaces in Exercises 6576. yl_x2_zl= 1
 12.12.76: Identify and sketch the surfaces in Exercises 6576. z2 _x2 _ y2 = I
Solutions for Chapter 12: Vectors and the Geometry of Space
Full solutions for Thomas' Calculus  12th Edition
ISBN: 9780321587992
Solutions for Chapter 12: Vectors and the Geometry of Space
Get Full SolutionsSince 76 problems in chapter 12: Vectors and the Geometry of Space have been answered, more than 8172 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Thomas' Calculus, edition: 12. This expansive textbook survival guide covers the following chapters and their solutions. Thomas' Calculus was written by and is associated to the ISBN: 9780321587992. Chapter 12: Vectors and the Geometry of Space includes 76 full stepbystep solutions.

Absolute maximum
A value ƒ(c) is an absolute maximum value of ƒ if ƒ(c) ? ƒ(x) for all x in the domain of ƒ.

Bounded above
A function is bounded above if there is a number B such that ƒ(x) ? B for all x in the domain of ƒ.

Circular functions
Trigonometric functions when applied to real numbers are circular functions

Combinatorics
A branch of mathematics related to determining the number of elements of a set or the number of ways objects can be arranged or combined

Cosecant
The function y = csc x

Direction vector for a line
A vector in the direction of a line in threedimensional space

Higherdegree polynomial function
A polynomial function whose degree is ? 3

Infinite limit
A special case of a limit that does not exist.

Interval notation
Notation used to specify intervals, pp. 4, 5.

Midpoint (on a number line)
For the line segment with endpoints a and b, a + b2

Multiplication principle of probability
If A and B are independent events, then P(A and B) = P(A) # P(B). If Adepends on B, then P(A and B) = P(AB) # P(B)

Normal distribution
A distribution of data shaped like the normal curve.

Polar coordinate system
A coordinate system whose ordered pair is based on the directed distance from a central point (the pole) and the angle measured from a ray from the pole (the polar axis)

Powerreducing identity
A trigonometric identity that reduces the power to which the trigonometric functions are raised.

Quadrantal angle
An angle in standard position whose terminal side lies on an axis.

Quotient identities
tan ?= sin ?cos ?and cot ?= cos ? sin ?

Rectangular coordinate system
See Cartesian coordinate system.

Tangent
The function y = tan x

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

Variation
See Power function.