 3.5.1: In Exercises 12, let u = (3, 2, 1), v = (0, 2, 3), and w = (2, 6, 7...
 3.5.2: In Exercises 12, let u = (3, 2, 1), v = (0, 2, 3), and w = (2, 6, 7...
 3.5.3: In Exercises 34, let u, v, and w be the vectors in Exercises 12. Us...
 3.5.4: In Exercises 34, let u, v, and w be the vectors in Exercises 12. Us...
 3.5.5: In Exercises 56, let u, v, and w be the vectors in Exercises 12. Co...
 3.5.6: In Exercises 56, let u, v, and w be the vectors in Exercises 12. Co...
 3.5.7: In Exercises 78, use the cross product to find a vector that is ort...
 3.5.8: In Exercises 78, use the cross product to find a vector that is ort...
 3.5.9: In Exercises 910, find the area of the parallelogram determined by ...
 3.5.10: In Exercises 910, find the area of the parallelogram determined by ...
 3.5.11: In Exercises 1112, find the area of the parallelogram with the give...
 3.5.12: In Exercises 1112, find the area of the parallelogram with the give...
 3.5.13: In Exercises 1314, find the area of the triangle with the given ver...
 3.5.14: In Exercises 1314, find the area of the triangle with the given ver...
 3.5.15: In Exercises 1516, find the area of the triangle in 3space that ha...
 3.5.16: In Exercises 1516, find the area of the triangle in 3space that ha...
 3.5.17: In Exercises 1718, find the volume of the parallelepiped with sides...
 3.5.18: In Exercises 1718, find the volume of the parallelepiped with sides...
 3.5.19: In Exercises 1920, determine whether u, v, and w lie in the same pl...
 3.5.20: In Exercises 1920, determine whether u, v, and w lie in the same pl...
 3.5.21: In Exercises 2124, compute the scalar triple product u (v w).u = (2...
 3.5.22: In Exercises 2124, compute the scalar triple product u (v w).u = (1...
 3.5.23: In Exercises 2124, compute the scalar triple product u (v w).u = (a...
 3.5.24: In Exercises 2124, compute the scalar triple product u (v w).u = i,...
 3.5.25: In Exercises 2526, suppose that u (v w) = 3. Find(a) u (w v) (b) (v...
 3.5.26: In Exercises 2526, suppose that u (v w) = 3. Find(a) v (u w) (b) (u...
 3.5.27: (a) Find the area of the triangle having vertices A(1, 0, 1), B(0, ...
 3.5.28: Use the cross product to find the sine of the angle between the vec...
 3.5.29: Simplify (u + v) (u v)
 3.5.30: Let a = (a1, a2, a3), b = (b1, b2, b3), c = (c1, c2, c3), and d = (...
 3.5.31: Exercises 3132 You know from your own experience that the tendency ...
 3.5.32: Exercises 3132 You know from your own experience that the tendency ...
 3.5.33: Let u, v, and w be nonzero vectors in 3space with the same initial...
 3.5.34: Prove the following identities. (a) (u + kv) v = u v (b) u (v z) = ...
 3.5.35: Prove: If a, b, c, and d lie in the same plane, then (a b) (c d) = 0
 3.5.36: Prove: If is the angle between u and v and u v = 0, then tan = u v ...
 3.5.37: Prove that if u, v, and w are vectors in R3, no two of which are co...
 3.5.38: It is a theorem of solid geometry that the volume of a tetrahedron ...
 3.5.39: Use the result of Exercise 38 to find the volume of the tetrahedron...
 3.5.40: Prove part (d) of Theorem 3.5.1. [Hint: First prove the result in t...
 3.5.41: Prove part (e) of Theorem 3.5.1. [Hint: Apply part (a) of Theorem 3...
 3.5.42: Prove: (a) Prove (b) of Theorem 3.5.2. (b) Prove (c) of Theorem 3.5...
Solutions for Chapter 3.5: Cross Product
Full solutions for Elementary Linear Algebra, Binder Ready Version: Applications Version  11th Edition
ISBN: 9781118474228
Solutions for Chapter 3.5: Cross Product
Get Full SolutionsChapter 3.5: Cross Product includes 42 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Elementary Linear Algebra, Binder Ready Version: Applications Version, edition: 11. Since 42 problems in chapter 3.5: Cross Product have been answered, more than 14947 students have viewed full stepbystep solutions from this chapter. Elementary Linear Algebra, Binder Ready Version: Applications Version was written by and is associated to the ISBN: 9781118474228.

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

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

Column space C (A) =
space of all combinations of the columns of A.

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.

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.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

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.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

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

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·