 12.1: Using a scale of 1 cm represents 10 units, sketch a vector to repre...
 12.2: If represents a velocity of 50 m s1 due east, draw a directed line ...
 12.3: Draw a scale diagram to represent the following vectors: a a force ...
 12.4: Consider: One way of finding !PS is: !PS = !PR + !RS = (a + b) + c....
 12.5: Answer the Opening page 276.
 12.6: For r = 2 3 and s = 1 4 find: a j r j b jsj c j r + sj d j r sj e j...
 12.7: If p = 1 3 and q = 2 4 find: a j p j b j 2p j c j2p j d j 3p j e j ...
 12.8: Suppose x = x1 x2 and a = a1 a2 . Show by equating components, that...
 12.9: From your answers in 7, you should have noticed that j kv j = j k j...
 12.10: The vertices of triangle ABC are A(5, 6, 2), B(6, 12, 9), and C(2, ...
 12.11: A sphere has centre C(1, 2, 4) and diameter [AB] where A is (2, 1, ...
 12.12: a State the coordinates of any general point A on the Y axis. b Us...
 12.13: Find a, b, and c if: a 0 @ a 4 b 3 c + 2 1 A = 0 @ 1 3 4 1 A b 0 @ ...
 12.14: Find k given the unit vector: a 0 B@ 1 2 k 1 4 1 CA b 0 B@ k 2 3 1 ...
 12.15: A(1, 3, 4), B(2, 5, 1), C(1, 2, 2), and D(r, s, t) are four points ...
 12.16: A quadrilateral has vertices A(1, 2, 3), B(3, 3, 2), C(7, 4, 5), an...
 12.17: PQRS is a parallelogram. P is (1, 2, 3), Q is (1, 2, 5), and R is (...
 12.18: Find the angle ABC of triangle ABC for A(3, 0, 1), B(3, 1, 2), and ...
 12.19: The cube alongside has sides of length 2 cm. Find, using vector met...
 12.20: [KL], [LM], and [LX] are 8, 5, and 3 units long respectively. P is ...
 12.21: Consider tetrahedron ABCD. a Find the coordinates of M. b Find the ...
 12.22: a Find t if 2i + tj + (t 2)k and ti + 3j + tk are perpendicular. b ...
 12.23: Find the angle made by: a i and 0 @ 1 2 3 1 A b j and 0 @ 1 1 3 1 A.
Solutions for Chapter 12: VECTORS AND SCALARS
Full solutions for Mathematics for the International Student: Mathematics SL  3rd Edition
ISBN: 9781921972089
Solutions for Chapter 12: VECTORS AND SCALARS
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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.

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

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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

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

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.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

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).

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.

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).

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

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

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

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·

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

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.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).