 10.1.10.1.1: Fill in the blanks. A _______ coordinate system can be formed by pa...
 10.1.10.1.2: Fill in the blanks. The three coordinate planes of a threedimensio...
 10.1.10.1.3: Fill in the blanks. The coordinate planes of a threedimensional co...
 10.1.10.1.4: Fill in the blanks. The distance between the points and can be foun...
 10.1.10.1.5: Fill in the blanks. The midpoint of the line segment joining the po...
 10.1.10.1.6: Fill in the blanks. A _______ is the set of all points such that th...
 10.1.10.1.7: Fill in the blanks.
 10.1.10.1.8: Fill in the blanks.
 10.1.10.1.9: In Exercises 510, plot each point in the same threedimensional coo...
 10.1.10.1.10: In Exercises 510, plot each point in the same threedimensional coo...
 10.1.10.1.11: In Exercises 1114, find the coordinates of the point.
 10.1.10.1.12: In Exercises 1114, find the coordinates of the point.
 10.1.10.1.13: In Exercises 1114, find the coordinates of the point.
 10.1.10.1.14: In Exercises 1114, find the coordinates of the point.
 10.1.10.1.15: In Exercises 1520, determine the octant(s) in which is located so t...
 10.1.10.1.16: In Exercises 1520, determine the octant(s) in which is located so t...
 10.1.10.1.17: In Exercises 1520, determine the octant(s) in which is located so t...
 10.1.10.1.18: In Exercises 1520, determine the octant(s) in which is located so t...
 10.1.10.1.19: In Exercises 1520, determine the octant(s) in which is located so t...
 10.1.10.1.20: In Exercises 1520, determine the octant(s) in which is located so t...
 10.1.10.1.21: In Exercises 2126, find the distance between the points.
 10.1.10.1.22: In Exercises 2126, find the distance between the points.
 10.1.10.1.23: In Exercises 2126, find the distance between the points.
 10.1.10.1.24: In Exercises 2126, find the distance between the points.
 10.1.10.1.25: In Exercises 2126, find the distance between the points.
 10.1.10.1.26: In Exercises 2126, find the distance between the points.
 10.1.10.1.27: In Exercises 2730, find the lengths of the sides of the right trian...
 10.1.10.1.28: In Exercises 2730, find the lengths of the sides of the right trian...
 10.1.10.1.29: In Exercises 2730, find the lengths of the sides of the right trian...
 10.1.10.1.30: In Exercises 2730, find the lengths of the sides of the right trian...
 10.1.10.1.31: In Exercises 3134, find the lengths of the sides of the triangle wi...
 10.1.10.1.32: In Exercises 3134, find the lengths of the sides of the triangle wi...
 10.1.10.1.33: In Exercises 3134, find the lengths of the sides of the triangle wi...
 10.1.10.1.34: In Exercises 3134, find the lengths of the sides of the triangle wi...
 10.1.10.1.35: In Exercises 35 40, find the midpoint of the line segment joining t...
 10.1.10.1.36: In Exercises 35 40, find the midpoint of the line segment joining t...
 10.1.10.1.37: In Exercises 35 40, find the midpoint of the line segment joining t...
 10.1.10.1.38: In Exercises 35 40, find the midpoint of the line segment joining t...
 10.1.10.1.39: In Exercises 35 40, find the midpoint of the line segment joining t...
 10.1.10.1.40: In Exercises 35 40, find the midpoint of the line segment joining t...
 10.1.10.1.41: In Exercises 4150, find the standard form of the equation of the sp...
 10.1.10.1.42: In Exercises 4150, find the standard form of the equation of the sp...
 10.1.10.1.43: In Exercises 4150, find the standard form of the equation of the sp...
 10.1.10.1.44: In Exercises 4150, find the standard form of the equation of the sp...
 10.1.10.1.45: In Exercises 4150, find the standard form of the equation of the sp...
 10.1.10.1.46: In Exercises 4150, find the standard form of the equation of the sp...
 10.1.10.1.47: In Exercises 4150, find the standard form of the equation of the sp...
 10.1.10.1.48: In Exercises 4150, find the standard form of the equation of the sp...
 10.1.10.1.49: In Exercises 4150, find the standard form of the equation of the sp...
 10.1.10.1.50: In Exercises 4150, find the standard form of the equation of the sp...
 10.1.10.1.51: In Exercises 5164, find the center and radius of the sphere.
 10.1.10.1.52: In Exercises 5164, find the center and radius of the sphere.
 10.1.10.1.53: In Exercises 5164, find the center and radius of the sphere.
 10.1.10.1.54: In Exercises 5164, find the center and radius of the sphere.
 10.1.10.1.55: In Exercises 5164, find the center and radius of the sphere.
 10.1.10.1.56: In Exercises 5164, find the center and radius of the sphere.
 10.1.10.1.57: In Exercises 5164, find the center and radius of the sphere.
 10.1.10.1.58: In Exercises 5164, find the center and radius of the sphere.
 10.1.10.1.59: In Exercises 5164, find the center and radius of the sphere.
 10.1.10.1.60: In Exercises 5164, find the center and radius of the sphere.
 10.1.10.1.61: In Exercises 5164, find the center and radius of the sphere.
 10.1.10.1.62: In Exercises 5164, find the center and radius of the sphere.
 10.1.10.1.63: In Exercises 5164, find the center and radius of the sphere.
 10.1.10.1.64: In Exercises 5164, find the center and radius of the sphere.
 10.1.10.1.65: In Exercises 6570, sketch the graph of the equation and sketch the ...
 10.1.10.1.66: In Exercises 6570, sketch the graph of the equation and sketch the ...
 10.1.10.1.67: In Exercises 6570, sketch the graph of the equation and sketch the ...
 10.1.10.1.68: In Exercises 6570, sketch the graph of the equation and sketch the ...
 10.1.10.1.69: In Exercises 6570, sketch the graph of the equation and sketch the ...
 10.1.10.1.70: In Exercises 6570, sketch the graph of the equation and sketch the ...
 10.1.10.1.71: In Exercises 7174, use a threedimensional graphing utility to grap...
 10.1.10.1.72: In Exercises 7174, use a threedimensional graphing utility to grap...
 10.1.10.1.73: In Exercises 7174, use a threedimensional graphing utility to grap...
 10.1.10.1.74: In Exercises 7174, use a threedimensional graphing utility to grap...
 10.1.10.1.75: Crystals Crystals are classified according to their symmetry. Cryst...
 10.1.10.1.76: Crystals Crystals shaped like rectangular prisms are classified as ...
 10.1.10.1.77: Architecture A spherical building has a diameter of 165 feet. The c...
 10.1.10.1.78: Geography Assume that Earth is a sphere with a radius of 3963 miles...
 10.1.10.1.79: True or False? In Exercises 79 and 80, determine whether the statem...
 10.1.10.1.80: True or False? In Exercises 79 and 80, determine whether the statem...
 10.1.10.1.81: Think About It What is the zcoordinate of any point in the xyplan...
 10.1.10.1.82: Writing In twodimensional coordinate geometry, the graph of the eq...
 10.1.10.1.83: A sphere intersects the yzplane. Describe the trace.
 10.1.10.1.84: A plane intersects the xyplane. Describe the trace
 10.1.10.1.85: A line segment has as one endpoint and as its midpoint. Find the ot...
 10.1.10.1.86: Use the result of Exercise 85 to find the coordinates of one endpoi...
 10.1.10.1.87: In Exercises 8792, solve the quadratic equation by completing the s...
 10.1.10.1.88: In Exercises 8792, solve the quadratic equation by completing the s...
 10.1.10.1.89: In Exercises 8792, solve the quadratic equation by completing the s...
 10.1.10.1.90: In Exercises 8792, solve the quadratic equation by completing the s...
 10.1.10.1.91: In Exercises 8792, solve the quadratic equation by completing the s...
 10.1.10.1.92: In Exercises 8792, solve the quadratic equation by completing the s...
 10.1.10.1.93: In Exercises 9396, find the magnitude and direction angle of the ve...
 10.1.10.1.94: In Exercises 9396, find the magnitude and direction angle of the ve...
 10.1.10.1.95: In Exercises 9396, find the magnitude and direction angle of the ve...
 10.1.10.1.96: In Exercises 9396, find the magnitude and direction angle of the ve...
 10.1.10.1.97: In Exercises 97 and 98, find the dot product of u and v.
 10.1.10.1.98: In Exercises 97 and 98, find the dot product of u and v.
 10.1.10.1.99: In Exercises 99102, write the first five terms of the sequence begi...
 10.1.10.1.100: In Exercises 99102, write the first five terms of the sequence begi...
 10.1.10.1.101: In Exercises 99102, write the first five terms of the sequence begi...
 10.1.10.1.102: In Exercises 99102, write the first five terms of the sequence begi...
 10.1.10.1.103: In Exercises 103110, find the standard form of the equation of the ...
 10.1.10.1.104: In Exercises 103110, find the standard form of the equation of the ...
 10.1.10.1.105: In Exercises 103110, find the standard form of the equation of the ...
 10.1.10.1.106: In Exercises 103110, find the standard form of the equation of the ...
 10.1.10.1.107: In Exercises 103110, find the standard form of the equation of the ...
 10.1.10.1.108: In Exercises 103110, find the standard form of the equation of the ...
 10.1.10.1.109: In Exercises 103110, find the standard form of the equation of the ...
 10.1.10.1.110: In Exercises 103110, find the standard form of the equation of the ...
Solutions for Chapter 10.1: The ThreeDimensional Coordinate System
Full solutions for Precalculus With Limits A Graphing Approach  5th Edition
ISBN: 9780618851522
Solutions for Chapter 10.1: The ThreeDimensional Coordinate System
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 110 problems in chapter 10.1: The ThreeDimensional Coordinate System have been answered, more than 45941 students have viewed full stepbystep solutions from this chapter. Precalculus With Limits A Graphing Approach was written by and is associated to the ISBN: 9780618851522. Chapter 10.1: The ThreeDimensional Coordinate System includes 110 full stepbystep solutions. This textbook survival guide was created for the textbook: Precalculus With Limits A Graphing Approach, edition: 5.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.