 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 3975 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 Sieva Kozinsky and is associated to the ISBN: 9780321587992. Chapter 12: Vectors and the Geometry of Space includes 76 full stepbystep solutions.

Additive inverse of a complex number
The opposite of a + bi, or a  bi

Arccosecant function
See Inverse cosecant function.

Binomial theorem
A theorem that gives an expansion formula for (a + b)n

Bounded interval
An interval that has finite length (does not extend to ? or ?)

Combination
An arrangement of elements of a set, in which order is not important

First octant
The points (x, y, z) in space with x > 0 y > 0, and z > 0.

Line of symmetry
A line over which a graph is the mirror image of itself

Mathematical model
A mathematical structure that approximates phenomena for the purpose of studying or predicting their behavior

Midpoint (in Cartesian space)
For the line segment with endpoints (x 1, y1, z 1) and (x2, y2, z2), ax 1 + x 22 ,y1 + y22 ,z 1 + z 22 b

Multiplication property of inequality
If u < v and c > 0, then uc < vc. If u < and c < 0, then uc > vc

Natural logarithm
A logarithm with base e.

Numerical derivative of ƒ at a
NDER f(a) = ƒ1a + 0.0012  ƒ1a  0.00120.002

Polynomial interpolation
The process of fitting a polynomial of degree n to (n + 1) points.

Present value of an annuity T
he net amount of your money put into an annuity.

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

Rose curve
A graph of a polar equation or r = a cos nu.

Row echelon form
A matrix in which rows consisting of all 0’s occur only at the bottom of the matrix, the first nonzero entry in any row with nonzero entries is 1, and the leading 1’s move to the right as we move down the rows.

Solution of an equation or inequality
A value of the variable (or values of the variables) for which the equation or inequality is true

Sum of an infinite geometric series
Sn = a 1  r , r 6 1

xaxis
Usually the horizontal coordinate line in a Cartesian coordinate system with positive direction to the right,.
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