 9.2.9.1.39: Let a = [2. I. 4]. b = [4, 0.3], c = [3, 2, 1]. Find
 9.2.9.1.40: Let a = [2. I. 4]. b = [4, 0.3], c = [3, 2, 1]. Find
 9.2.9.1.41: Let a = [2. I. 4]. b = [4, 0.3], c = [3, 2, 1]. Find
 9.2.9.1.42: Let a = [2. I. 4]. b = [4, 0.3], c = [3, 2, 1]. Find
 9.2.9.1.43: Let a = [2. I. 4]. b = [4, 0.3], c = [3, 2, 1]. Find
 9.2.9.1.44: Let a = [2. I. 4]. b = [4, 0.3], c = [3, 2, 1]. Find
 9.2.9.1.45: Let a = [2. I. 4]. b = [4, 0.3], c = [3, 2, 1]. Find
 9.2.9.1.46: Let a = [2. I. 4]. b = [4, 0.3], c = [3, 2, 1]. Find
 9.2.9.1.47: Let a = [2. I. 4]. b = [4, 0.3], c = [3, 2, 1]. Find
 9.2.9.1.48: Let a = [2. I. 4]. b = [4, 0.3], c = [3, 2, 1]. Find
 9.2.9.1.49: Let a = [2. I. 4]. b = [4, 0.3], c = [3, 2, 1]. Find
 9.2.9.1.50: Let a = [2. I. 4]. b = [4, 0.3], c = [3, 2, 1]. Find
 9.2.9.1.51: What laws do Probs. I. 3.4, 7, 8 illustrate?
 9.2.9.1.52: Does uov = uow with u 0/= 0 imply that v = w?
 9.2.9.1.53: rove the CauchySchwarz inequality.
 9.2.9.1.54: Verify the CauchySchwarz inequality, the triangle inequality. and ...
 9.2.9.1.55: Prove the parallelogram equality.
 9.2.9.1.56: (Triangle inequality) Prove (7). Hint. Use (3) for la + bl and (6) ...
 9.2.9.1.57: Find the work done by a force p acting on a body if the body is dis...
 9.2.9.1.58: Find the work done by a force p acting on a body if the body is dis...
 9.2.9.1.59: Find the work done by a force p acting on a body if the body is dis...
 9.2.9.1.60: Find the work done by a force p acting on a body if the body is dis...
 9.2.9.1.61: Why is the work in Prob. 19 zero? Can work be negative? Explain.
 9.2.9.1.62: Show that the work done by the resultant of p and q in a displaceme...
 9.2.9.1.63: Find the work W = pod if d = 2i and p = i, i + j, j, i + j and ske...
 9.2.9.1.64: Let a = ll, I, I], b = [2.3.1], c = [I, 1,0]. Find the angle betwe...
 9.2.9.1.65: Let a = ll, I, I], b = [2.3.1], c = [I, 1,0]. Find the angle betwe...
 9.2.9.1.66: Let a = ll, I, I], b = [2.3.1], c = [I, 1,0]. Find the angle betwe...
 9.2.9.1.67: Let a = ll, I, I], b = [2.3.1], c = [I, 1,0]. Find the angle betwe...
 9.2.9.1.68: Let a = ll, I, I], b = [2.3.1], c = [I, 1,0]. Find the angle betwe...
 9.2.9.1.69: (Planes) Find the angle between the planes x + y + .;; = 1 and 2x ...
 9.2.9.1.70: (Cosine law) Deduce the law of cosines by using vectors a, b, and ab.
 9.2.9.1.71: (Triangle) Find the angles of the triangle with vertices [0, 0, 0],...
 9.2.9.1.72: (Addition law) Obtain cos (a  (3) = cos a cos {3 + sin a sin {3 by...
 9.2.9.1.73: (Parallelogram) Find the angles if the sides are [5, 0] and [I. 21
 9.2.9.1.74: (Distance) Find the distance of the plane 5x + 2y + z = 10 from the...
 9.2.9.1.75: Find the component of a in the direction of b. a = [I. 1,3]. b = [0...
 9.2.9.1.76: Find the component of a in the direction of b.a = [2. O. 6], b = [3...
 9.2.9.1.77: Find the component of a in the direction of b. a = [0.4, 3]. b = [...
 9.2.9.1.78: Find the component of a in the direction of b. a = [1,2,0]. b = [1...
 9.2.9.1.79: Under what condition will the projection of a in the direction of b...
 9.2.9.1.80: coordinates. such as Cartesian coordinates, whose "natural basis" (...
Solutions for Chapter 9.2: Inner Product (Dot Product)
Full solutions for Advanced Engineering Mathematics  9th Edition
ISBN: 9780471488859
Solutions for Chapter 9.2: Inner Product (Dot Product)
Get Full SolutionsChapter 9.2: Inner Product (Dot Product) includes 42 full stepbystep solutions. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9780471488859. Since 42 problems in chapter 9.2: Inner Product (Dot Product) have been answered, more than 46043 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 9.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

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

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

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.

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

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

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

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.

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

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.

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

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

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).

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

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

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

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).