 10.1: Plot and then label the points , , , , and .
 10.2: Give the coordinates of each point A, B, C, D, and E. Also name the...
 10.3: Find the distance between each pair of points: a) and (5, 1) c) (0,...
 10.4: If the distance between and is 5 units, find all possible values of a.
 10.5: If the distance between (b, 3) and (7, 3) is 3.5 units, find all po...
 10.6: Find an expression for the distance between (a, b) and (a, c) if .
 10.7: Find the distance between each pair of points: a) and (4, 0) c) (3,...
 10.8: Find the distance between each pair of points: a) and (2, 5) c) and...
 10.9: Find the midpoint of the line segment that joins each pair of point...
 10.10: Find the midpoint of the line segment that joins each pair of point...
 10.11: Points A and B have symmetry with respect to the origin O. Find the...
 10.12: Points A and B have symmetry with respect to point C(2, 3). Find th...
 10.13: Points A and B have symmetry with respect to point C. Find the coor...
 10.14: Points A and B have symmetry with respect to the x axis. Find the c...
 10.15: Points A and B have symmetry with respect to the x axis. Find the c...
 10.16: Points A and B have symmetry with respect to the vertical line wher...
 10.17: Points A and B have symmetry with respect to the y axis. Find the c...
 10.18: Points A and B have symmetry with respect to either the x axis or t...
 10.19: Points A and B have symmetry with respect to a vertical line or a h...
 10.20: In Exercises 20 to 22, apply the Midpoint Formula. is the midpoint ...
 10.21: In Exercises 20 to 22, apply the Midpoint Formula. is the midpoint ...
 10.22: In Exercises 20 to 22, apply the Midpoint Formula. A circle has its...
 10.23: A rectangle ABCD has three of its vertices at , , and C(6, 3). Find...
 10.24: A rectangle MNPQ has three of its vertices at M(0, 0), N(a, 0), and...
 10.25: Use the Distance Formula to determine the type of triangle that has...
 10.26: Use the method of Example 4 to find the equation of the line that d...
 10.27: Use the method of Example 4 to find the equation of the line that d...
 10.28: For coplanar points A, B, and C, suppose that you have used the Dis...
 10.29: If two vertices of an equilateral triangle are at (0, 0) and , what...
 10.30: The rectangle whose vertices are A(0, 0), B(a, 0), C(a, b), and D(0...
 10.31: There are two points on the y axis that are located a distance of 6...
 10.32: There are two points on the x axis that are located a distance of 6...
 10.33: The triangle that has vertices at , , and Q(2, 4) has been boxed in...
 10.34: Use the method suggested in Exercise 33 to find the area of , with ...
 10.35: Determine the area of if , , and C is the reflection of B across th...
 10.36: Find the area of in Exercise 35, but assume that C is the reflectio...
 10.37: For Exercises 37 to 42, refer to formulas for Chapter 9. Find the e...
 10.38: For Exercises 37 to 42, refer to formulas for Chapter 9. Find the e...
 10.39: For Exercises 37 to 42, refer to formulas for Chapter 9. Find the e...
 10.40: For Exercises 37 to 42, refer to formulas for Chapter 9. Find the e...
 10.41: For Exercises 37 to 42, refer to formulas for Chapter 9. Find the e...
 10.42: For Exercises 37 to 42, refer to formulas for Chapter 9. Find the v...
 10.43: By definition, an ellipse is the locus of points whose sum of dista...
 10.44: By definition, a hyperbola is the locus of points whose positive di...
 10.45: Use the Distance Formula to show that the equation of the parabola ...
 10.46: Use the Distance Formula to show that the equation of the parabola ...
 10.47: Following a 90 counterclockwise rotation about the origin, the imag...
 10.48: Consider the point . What is the image of C after a counterclockwis...
 10.49: Given the point , find the image of D after a counterclockwise rota...
 10.50: Determine the point of intersection, if such a point exists, for th...
Solutions for Chapter 10: Analytic Geometry
Full solutions for Elementary Geometry for College Students  6th Edition
ISBN: 9781285195698
Solutions for Chapter 10: Analytic Geometry
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Elementary Geometry for College Students, edition: 6. Elementary Geometry for College Students was written by Patricia and is associated to the ISBN: 9781285195698. Since 50 problems in chapter 10: Analytic Geometry have been answered, more than 1427 students have viewed full stepbystep solutions from this chapter. Chapter 10: Analytic Geometry includes 50 full stepbystep solutions.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

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

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

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.

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

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

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

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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

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

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.
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