 10.4.10.4.1: Fill in the blanks. The _______ vector for a line is given by _____...
 10.4.10.4.2: Fill in the blanks. The _______ of a line in space are given by a
 10.4.10.4.3: Fill in the blanks. If the direction numbers and of the vector are ...
 10.4.10.4.4: Fill in the blanks. A vector that is perpendicular to a plane is ca...
 10.4.10.4.5: Fill in the blanks. The standard form of the equation of a plane is...
 10.4.10.4.6: In Exercises 16, find (a) a set of parametric equations and (b) if ...
 10.4.10.4.7: In Exercises 714, find (a) a set of parametric equations and (b) if...
 10.4.10.4.8: In Exercises 714, find (a) a set of parametric equations and (b) if...
 10.4.10.4.9: In Exercises 714, find (a) a set of parametric equations and (b) if...
 10.4.10.4.10: In Exercises 714, find (a) a set of parametric equations and (b) if...
 10.4.10.4.11: In Exercises 714, find (a) a set of parametric equations and (b) if...
 10.4.10.4.12: In Exercises 714, find (a) a set of parametric equations and (b) if...
 10.4.10.4.13: In Exercises 714, find (a) a set of parametric equations and (b) if...
 10.4.10.4.14: In Exercises 714, find (a) a set of parametric equations and (b) if...
 10.4.10.4.15: In Exercises 15 and 16, sketch a graph of the line.
 10.4.10.4.16: In Exercises 15 and 16, sketch a graph of the line.
 10.4.10.4.17: In Exercises 1722, find the general form of the equation of the pla...
 10.4.10.4.18: In Exercises 1722, find the general form of the equation of the pla...
 10.4.10.4.19: In Exercises 1722, find the general form of the equation of the pla...
 10.4.10.4.20: In Exercises 1722, find the general form of the equation of the pla...
 10.4.10.4.21: In Exercises 1722, find the general form of the equation of the pla...
 10.4.10.4.22: In Exercises 1722, find the general form of the equation of the pla...
 10.4.10.4.23: In Exercises 2326, find the general form of the equation of the pla...
 10.4.10.4.24: In Exercises 2326, find the general form of the equation of the pla...
 10.4.10.4.25: In Exercises 2326, find the general form of the equation of the pla...
 10.4.10.4.26: In Exercises 2326, find the general form of the equation of the pla...
 10.4.10.4.27: In Exercises 2732, find the general form of the equation of the pla...
 10.4.10.4.28: In Exercises 2732, find the general form of the equation of the pla...
 10.4.10.4.29: In Exercises 2732, find the general form of the equation of the pla...
 10.4.10.4.30: In Exercises 2732, find the general form of the equation of the pla...
 10.4.10.4.31: In Exercises 2732, find the general form of the equation of the pla...
 10.4.10.4.32: In Exercises 2732, find the general form of the equation of the pla...
 10.4.10.4.33: In Exercises 3340, find a set of parametric equations of the line. ...
 10.4.10.4.34: In Exercises 3340, find a set of parametric equations of the line. ...
 10.4.10.4.35: In Exercises 3340, find a set of parametric equations of the line. ...
 10.4.10.4.36: In Exercises 3340, find a set of parametric equations of the line. ...
 10.4.10.4.37: In Exercises 3340, find a set of parametric equations of the line. ...
 10.4.10.4.38: In Exercises 3340, find a set of parametric equations of the line. ...
 10.4.10.4.39: In Exercises 3340, find a set of parametric equations of the line. ...
 10.4.10.4.40: In Exercises 3340, find a set of parametric equations of the line. ...
 10.4.10.4.41: In Exercises 41 44, determine whether the planes are parallel, orth...
 10.4.10.4.42: In Exercises 41 44, determine whether the planes are parallel, orth...
 10.4.10.4.43: In Exercises 41 44, determine whether the planes are parallel, orth...
 10.4.10.4.44: In Exercises 41 44, determine whether the planes are parallel, orth...
 10.4.10.4.45: In Exercises 4548, (a) find the angle between the two planes and (b...
 10.4.10.4.46: In Exercises 4548, (a) find the angle between the two planes and (b...
 10.4.10.4.47: In Exercises 4548, (a) find the angle between the two planes and (b...
 10.4.10.4.48: In Exercises 4548, (a) find the angle between the two planes and (b...
 10.4.10.4.49: In Exercises 4954, plot the intercepts and sketch a graph of the pl...
 10.4.10.4.50: In Exercises 4954, plot the intercepts and sketch a graph of the pl...
 10.4.10.4.51: In Exercises 4954, plot the intercepts and sketch a graph of the pl...
 10.4.10.4.52: In Exercises 4954, plot the intercepts and sketch a graph of the pl...
 10.4.10.4.53: In Exercises 4954, plot the intercepts and sketch a graph of the pl...
 10.4.10.4.54: In Exercises 4954, plot the intercepts and sketch a graph of the pl...
 10.4.10.4.55: In Exercises 5558, find the distance between the point and the plane.
 10.4.10.4.56: In Exercises 5558, find the distance between the point and the plane.
 10.4.10.4.57: In Exercises 5558, find the distance between the point and the plane.
 10.4.10.4.58: In Exercises 5558, find the distance between the point and the plane.
 10.4.10.4.59: Machine Design A tractor fuel tank has the shape and dimensions sho...
 10.4.10.4.60: Mechanical Design A chute at the top of a grain elevator of a combi...
 10.4.10.4.61: True or False? In Exercises 61 and 62, determine whether the statem...
 10.4.10.4.62: True or False? In Exercises 61 and 62, determine whether the statem...
 10.4.10.4.63: The direction numbers of two distinct lines in space are 10, 20, an...
 10.4.10.4.64: Exploration (a) Describe and find an equation for the surface gener...
 10.4.10.4.65: In Exercises 6568, convert the polar equation to rectangular form.
 10.4.10.4.66: In Exercises 6568, convert the polar equation to rectangular form.
 10.4.10.4.67: In Exercises 6568, convert the polar equation to rectangular form.
 10.4.10.4.68: In Exercises 6568, convert the polar equation to rectangular form.
 10.4.10.4.69: In Exercises 6972, convert the rectangular equation to polar form.
 10.4.10.4.70: In Exercises 6972, convert the rectangular equation to polar form.
 10.4.10.4.71: In Exercises 6972, convert the rectangular equation to polar form.
 10.4.10.4.72: In Exercises 6972, convert the rectangular equation to polar form.
Solutions for Chapter 10.4: Lines and Planes in Space
Full solutions for Precalculus With Limits A Graphing Approach  5th Edition
ISBN: 9780618851522
Solutions for Chapter 10.4: Lines and Planes in Space
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 72 problems in chapter 10.4: Lines and Planes in Space have been answered, more than 44102 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Precalculus With Limits A Graphing Approach, edition: 5. Precalculus With Limits A Graphing Approach was written by and is associated to the ISBN: 9780618851522. Chapter 10.4: Lines and Planes in Space includes 72 full stepbystep solutions.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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  ).

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

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