 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 2374 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.

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

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

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

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

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

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } 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.

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

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

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

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

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!

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.