 7.6.7.1.215: Find each of the following dot products. (6, 6)  (3, S)
 7.6.7.1.216: Find each of the following dot products. (3,4)  (S, S)
 7.6.7.1.217: Find each of the following dot products. (23,4)  (IS, 6)
 7.6.7.1.218: Find each of the following dot products. (11, S)  (4, 7)
 7.6.7.1.219: For each pair of vectors, find U  V.U = i + j, V = i  j
 7.6.7.1.220: For each pair of vectors, find U  V.U = i + j, V = i + j
 7.6.7.1.221: For each pair of vectors, find U  V.U = 6i, V = Sj
 7.6.7.1.222: For each pair of vectors, find U  V.U =  3i, V = Sj
 7.6.7.1.223: For each pair of vectors, find U  V.U = 2i +5j, V = Si + 2j
 7.6.7.1.224: For each pair of vectors, find U  V.U = Si + 3j, V = 3i + Sj
 7.6.7.1.225: Find the angle f) between the given vectors to the nearest tenth of...
 7.6.7.1.226: Find the angle f) between the given vectors to the nearest tenth of...
 7.6.7.1.227: Find the angle f) between the given vectors to the nearest tenth of...
 7.6.7.1.228: Find the angle f) between the given vectors to the nearest tenth of...
 7.6.7.1.229: Find the angle f) between the given vectors to the nearest tenth of...
 7.6.7.1.230: Find the angle f) between the given vectors to the nearest tenth of...
 7.6.7.1.231: Show that each pair of vectors is perpendicUlar. i andj
 7.6.7.1.232: Show that each pair of vectors is perpendicUlar. i + j and i  j
 7.6.7.1.233: Show that each pair of vectors is perpendicUlar. i and j
 7.6.7.1.234: Show that each pair of vectors is perpendicUlar. 2i + j and i  2j
 7.6.7.1.235: In general, show that the vectors V ai + hj and W = hi + aj are al...
 7.6.7.1.236: Find the value of a so that vectors U ai + 6j and V = 9i + 12j are ...
 7.6.7.1.237: Find the work perfonned when the given force F is applied to an obj...
 7.6.7.1.238: Find the work perfonned when the given force F is applied to an obj...
 7.6.7.1.239: Find the work perfonned when the given force F is applied to an obj...
 7.6.7.1.240: Find the work perfonned when the given force F is applied to an obj...
 7.6.7.1.241: Find the work perfonned when the given force F is applied to an obj...
 7.6.7.1.242: Find the work perfonned when the given force F is applied to an obj...
 7.6.7.1.243: Find the work perfonned when the given force F is applied to an obj...
 7.6.7.1.244: Find the work perfonned when the given force F is applied to an obj...
 7.6.7.1.245: Use the diagram shown in Figure 4 along with the law of cosines to ...
 7.6.7.1.246: Use Theorem 7.1 to prove Theorem 7.2.
 7.6.7.1.247: Work A package is pushed across a floor a distance of 75 feet by ex...
 7.6.7.1.248: Work A package is pushed across a floor a distance of 52 feet by ex...
 7.6.7.1.249: Work An automobile is pushed down a level street by exerting a forc...
 7.6.7.1.250: Work Mark pulls Allison and Mattie in a wagon by exerting a force o...
Solutions for Chapter 7.6: Triangles
Full solutions for Trigonometry
ISBN: 9780495108351
Solutions for Chapter 7.6: Triangles
Get Full SolutionsThis textbook survival guide was created for the textbook: Trigonometry, edition: . This expansive textbook survival guide covers the following chapters and their solutions. Since 36 problems in chapter 7.6: Triangles have been answered, more than 29880 students have viewed full stepbystep solutions from this chapter. Trigonometry was written by and is associated to the ISBN: 9780495108351. Chapter 7.6: Triangles includes 36 full stepbystep solutions.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

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.

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

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 II· Condition numbers measure the sensitivity of the output to change in the input.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = 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 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.

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

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).

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

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).

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 ().

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.