 13.1: An athlete can normally run with constant speed 6 m s1. Using a vec...
 13.2: In still water, Mary can swim at 1:2 m s1. She is standing at point...
 13.3: A boat needs to travel south at a speed of 20 km h1. However a cons...
 13.4: As part of an endurance race, Stephanie needs to swim from X to Y a...
 13.5: An aeroplane needs to fly due east from one city to another at a sp...
 13.6: Find the vector equation of the line which passes through: a A(1, 2...
 13.7: Find the coordinates of the point where the line with parametric eq...
 13.8: Find points on the line with parametric equations x = 2 t, y = 3+2t...
 13.9: A helicopter at A(6, 9, 3) moves with constant velocity in a straig...
Solutions for Chapter 13: PROBLEMS INVOLVING VECTOR OPERATIONS
Full solutions for Mathematics for the International Student: Mathematics SL  3rd Edition
ISBN: 9781921972089
Solutions for Chapter 13: PROBLEMS INVOLVING VECTOR OPERATIONS
Get Full SolutionsThis textbook survival guide was created for the textbook: Mathematics for the International Student: Mathematics SL, edition: 3. Chapter 13: PROBLEMS INVOLVING VECTOR OPERATIONS includes 9 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Mathematics for the International Student: Mathematics SL was written by and is associated to the ISBN: 9781921972089. Since 9 problems in chapter 13: PROBLEMS INVOLVING VECTOR OPERATIONS have been answered, more than 12403 students have viewed full stepbystep solutions from this chapter.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

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 columns of A.
Columns without pivots; these are combinations of earlier columns.

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.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

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

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).

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

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

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

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

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.