 9.6.1: The distance d from P1 = 1x1 , y12 to P2 = 1x2 , y22 is d = _______
 9.6.2: In space, points of the form 1x, y, 02 lie in a plane called the .
 9.6.3: If v = ai + bj + ck is a vector in space, the scalars a, b, c are c...
 9.6.4: The sum of the squares of the direction cosines of a vector in spac...
 9.6.5: True or False In space, the dot product of two vectors is a positiv...
 9.6.6: True or False A vector in space may be described by specifying its ...
 9.6.7: In 714, describe the set of points 1x, y, z2 defined by the equatio...
 9.6.8: In 714, describe the set of points 1x, y, z2 defined by the equatio...
 9.6.9: In 714, describe the set of points 1x, y, z2 defined by the equatio...
 9.6.10: In 714, describe the set of points 1x, y, z2 defined by the equatio...
 9.6.11: In 714, describe the set of points 1x, y, z2 defined by the equatio...
 9.6.12: In 714, describe the set of points 1x, y, z2 defined by the equatio...
 9.6.13: In 714, describe the set of points 1x, y, z2 defined by the equatio...
 9.6.14: In 714, describe the set of points 1x, y, z2 defined by the equatio...
 9.6.15: In 1520, find the distance from P1 to P2 .
 9.6.16: In 1520, find the distance from P1 to P2 .
 9.6.17: In 1520, find the distance from P1 to P2 .
 9.6.18: In 1520, find the distance from P1 to P2 .
 9.6.19: In 1520, find the distance from P1 to P2 .
 9.6.20: In 1520, find the distance from P1 to P2 .
 9.6.21: In 2126, opposite vertices of a rectangular box whose edges are par...
 9.6.22: In 2126, opposite vertices of a rectangular box whose edges are par...
 9.6.23: In 2126, opposite vertices of a rectangular box whose edges are par...
 9.6.24: In 2126, opposite vertices of a rectangular box whose edges are par...
 9.6.25: In 2126, opposite vertices of a rectangular box whose edges are par...
 9.6.26: In 2126, opposite vertices of a rectangular box whose edges are par...
 9.6.27: In 2732, the vector v has initial point P and terminal point Q. Wri...
 9.6.28: In 2732, the vector v has initial point P and terminal point Q. Wri...
 9.6.29: In 2732, the vector v has initial point P and terminal point Q. Wri...
 9.6.30: In 2732, the vector v has initial point P and terminal point Q. Wri...
 9.6.31: In 2732, the vector v has initial point P and terminal point Q. Wri...
 9.6.32: In 2732, the vector v has initial point P and terminal point Q. Wri...
 9.6.33: In 3338, find v.
 9.6.34: In 3338, find v.
 9.6.35: In 3338, find v.
 9.6.36: In 3338, find v.
 9.6.37: In 3338, find v.
 9.6.38: In 3338, find v.
 9.6.39: In 3944, find each quantity if v = 3i  5j + 2k and w = 2i + 3j  2k
 9.6.40: In 3944, find each quantity if v = 3i  5j + 2k and w = 2i + 3j  2k
 9.6.41: In 3944, find each quantity if v = 3i  5j + 2k and w = 2i + 3j  2k
 9.6.42: In 3944, find each quantity if v = 3i  5j + 2k and w = 2i + 3j  2k
 9.6.43: In 3944, find each quantity if v = 3i  5j + 2k and w = 2i + 3j  2k
 9.6.44: In 3944, find each quantity if v = 3i  5j + 2k and w = 2i + 3j  2k
 9.6.45: In 4550, find the unit vector in the same direction as v.
 9.6.46: In 4550, find the unit vector in the same direction as v.
 9.6.47: In 4550, find the unit vector in the same direction as v.
 9.6.48: In 4550, find the unit vector in the same direction as v.
 9.6.49: In 4550, find the unit vector in the same direction as v.
 9.6.50: In 4550, find the unit vector in the same direction as v.
 9.6.51: In 5158, find the dot product v ~ w and the angle between v and w.
 9.6.52: In 5158, find the dot product v ~ w and the angle between v and w.
 9.6.53: In 5158, find the dot product v ~ w and the angle between v and w.
 9.6.54: In 5158, find the dot product v ~ w and the angle between v and w.
 9.6.55: In 5158, find the dot product v ~ w and the angle between v and w.
 9.6.56: In 5158, find the dot product v ~ w and the angle between v and w.
 9.6.57: In 5158, find the dot product v ~ w and the angle between v and w.
 9.6.58: In 5158, find the dot product v ~ w and the angle between v and w.
 9.6.59: In 59 66, find the direction angles of each vector. Write each vect...
 9.6.60: In 59 66, find the direction angles of each vector. Write each vect...
 9.6.61: In 59 66, find the direction angles of each vector. Write each vect...
 9.6.62: In 59 66, find the direction angles of each vector. Write each vect...
 9.6.63: In 59 66, find the direction angles of each vector. Write each vect...
 9.6.64: In 59 66, find the direction angles of each vector. Write each vect...
 9.6.65: In 59 66, find the direction angles of each vector. Write each vect...
 9.6.66: In 59 66, find the direction angles of each vector. Write each vect...
 9.6.67: Consider the doublejointed robotic arm shown in the figure. Let th...
 9.6.68: In space, the collection of all points that are the same distance f...
 9.6.69: In 69 and 70, find the equation of a sphere with radius r and cente...
 9.6.70: In 69 and 70, find the equation of a sphere with radius r and cente...
 9.6.71: In 7176, find the radius and center of each sphere. [Hint: Complete...
 9.6.72: In 7176, find the radius and center of each sphere. [Hint: Complete...
 9.6.73: In 7176, find the radius and center of each sphere. [Hint: Complete...
 9.6.74: In 7176, find the radius and center of each sphere. [Hint: Complete...
 9.6.75: In 7176, find the radius and center of each sphere. [Hint: Complete...
 9.6.76: In 7176, find the radius and center of each sphere. [Hint: Complete...
 9.6.77: Find the work done by a force of 3 newtons acting in the direction ...
 9.6.78: Find the work done by a force of 1 newton acting in the direction 2...
 9.6.79: Find the work done in moving an object along a vector u = 3i + 2j ...
Solutions for Chapter 9.6: Vectors in Space
Full solutions for Precalculus Enhanced with Graphing Utilities  6th Edition
ISBN: 9780132854351
Solutions for Chapter 9.6: Vectors in Space
Get Full SolutionsChapter 9.6: Vectors in Space includes 79 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Precalculus Enhanced with Graphing Utilities, edition: 6. Since 79 problems in chapter 9.6: Vectors in Space have been answered, more than 56109 students have viewed full stepbystep solutions from this chapter. Precalculus Enhanced with Graphing Utilities was written by and is associated to the ISBN: 9780132854351.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

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

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).