 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 and is associated to the ISBN: 9781285195698. Since 50 problems in chapter 10: Analytic Geometry have been answered, more than 4222 students have viewed full stepbystep solutions from this chapter. Chapter 10: Analytic Geometry includes 50 full stepbystep solutions.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

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.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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!

Solvable system Ax = b.
The right side b is in the column space of A.

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.