 123.1: Tell whether each transformation appears to be a rotation.1
 123.2: Tell whether each transformation appears to be a rotation.2
 123.3: Tell whether each transformation appears to be a rotation.3
 123.4: Tell whether each transformation appears to be a rotation.4
 123.5: Copy each figure and the angle of rotation. Draw the rotation of th...
 123.6: Copy each figure and the angle of rotation. Draw the rotation of th...
 123.7: Rotate the figure with the given vertices about the origin using th...
 123.8: Rotate the figure with the given vertices about the origin using th...
 123.9: Rotate the figure with the given vertices about the origin using th...
 123.10: Rotate the figure with the given vertices about the origin using th...
 123.11: Animation An artist uses a coordinate plane to plan the motion ofan...
 123.12: Tell whether each transformation appears to be a rotation.12
 123.13: Tell whether each transformation appears to be a rotation.13
 123.14: Tell whether each transformation appears to be a rotation.14
 123.15: Tell whether each transformation appears to be a rotation.15
 123.16: Copy each figure and the angle of rotation. Draw the rotation of th...
 123.17: Copy each figure and the angle of rotation. Draw the rotation of th...
 123.18: Rotate the figure with the given vertices about the origin using th...
 123.19: Rotate the figure with the given vertices about the origin using th...
 123.20: Rotate the figure with the given vertices about the origin using th...
 123.21: Rotate the figure with the given vertices about the origin using th...
 123.22: Architecture The CN Tower in Toronto, Canada, features a revolving ...
 123.23: Copy each figure. Then draw the rotation of the figure about the re...
 123.24: Copy each figure. Then draw the rotation of the figure about the re...
 123.25: Copy each figure. Then draw the rotation of the figure about the re...
 123.26: Point Q has coordinates (2, 3) . After a rotation about the origin,...
 123.27: Rectangle RSTU is the image of rectangle LMNP under a180 rotation a...
 123.28: Rectangle RSTU is the image of rectangle LMNP under a180 rotation a...
 123.29: Rectangle RSTU is the image of rectangle LMNP under a180 rotation a...
 123.30: Rectangle RSTU is the image of rectangle LMNP under a180 rotation a...
 123.31: This problem will prepare you for the ConceptConnection on page 854...
 123.32: Each figure shows a preimage and its image under a rotation. Copy t...
 123.33: Each figure shows a preimage and its image under a rotation. Copy t...
 123.34: Each figure shows a preimage and its image under a rotation. Copy t...
 123.35: Astronomy The photograph was made byplacing a camera on a tripod an...
 123.36: Estimation In the diagram, ABC A'B'C' undera rotation about point P...
 123.37: Critical Thinking A student wrote the following in his mathjournal....
 123.38: Use the figure for Exercises 3840.Sketch the image of pentagon ABCD...
 123.39: Use the figure for Exercises 3840.Sketch the image of pentagon ABCD...
 123.40: Use the figure for Exercises 3840.Write About It Is the image of AB...
 123.41: Construction Copy the figure. Use the construction of an angle cong...
 123.42: What is the image of the point (2, 5) when it is rotated about the...
 123.43: The six cars of a Ferris wheel are located at the vertices of a reg...
 123.44: Gridded Response Under a rotation about the origin,the point (3, 4...
 123.45: Engineering Gears are used to changethe speed and direction of rota...
 123.46: Given: A'B' is the rotation image of AB about point P. Prove: AB A'...
 123.47: Once you have proved that the rotation image of a segment iscongrue...
 123.48: Once you have proved that the rotation image of a segment iscongrue...
 123.49: Once you have proved that the rotation image of a segment iscongrue...
 123.50: Once you have proved that the rotation image of a segment iscongrue...
 123.51: Once you have proved that the rotation image of a segment iscongrue...
 123.52: Find the value(s) of x when y is 3. (Previous course)y = x 2  4x + 7
 123.53: Find the value(s) of x when y is 3. (Previous course)y = 2x 2  5x  9
 123.54: Find the value(s) of x when y is 3. (Previous course)y = x 2  2
 123.55: Find each measure. (Lesson 66) mXYR
 123.56: Find each measure. (Lesson 66) QR
 123.57: iven the points A (1, 3) , B (5, 0) , C (3, 2) , and D (5, 6) ,f...
 123.58: iven the points A (1, 3) , B (5, 0) , C (3, 2) , and D (5, 6) ,f...
 123.59: iven the points A (1, 3) , B (5, 0) , C (3, 2) , and D (5, 6) ,f...
Solutions for Chapter 123: Rotations
Full solutions for Geometry  1st Edition
ISBN: 9780030923456
Solutions for Chapter 123: Rotations
Get Full SolutionsGeometry was written by and is associated to the ISBN: 9780030923456. Chapter 123: Rotations includes 59 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 59 problems in chapter 123: Rotations have been answered, more than 43714 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Geometry, edition: 1.

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

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib IIĀ· Condition numbers measure the sensitivity of the output to change in the input.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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

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.

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!).

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

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.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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).

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

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

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

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