- 9.1: Find the area of:
- 9.2: If triangle ABC has area 150 cm2, find the value of x:
- 9.3: A golfer played his tee shot a distance of 220 m to point A. He the...
- 9.4: A communications tower is constructed on top of a building as shown...
- 9.5: Hikers Ritva and Esko leave point P at the same time. Ritva walks 4...
- 9.6: A football goal is 5 metres wide. When a player is 26 metres from o...
- 9.7: A tower 42 metres high stands on top of a hill. From a point some d...
- 9.8: From the foot of a building I have to look 22 upwards to sight the ...
- 9.9: Find the measure of PQR in the rectangular b box shown.
- 9.10: Two observation posts A and B are 12 km apart. A third observation ...
- 9.11: Stan and Olga are considering buying a sheep farm. A surveyor has s...
- 9.12: Thabo and Palesa start at point A. They each walk in a straight lin...
- 9.13: The cross-section design of the kerbing for a driverless-bus roadwa...
- 9.14: An orienteer runs for 4 1 2 km, then turns through an angle of 32 a...
- 9.15: Sam and Markus are standing on level ground 100 metres apart. A lar...
- 9.16: A helicopter A observes two ships B and C. B is 23:8 km from the he...
Solutions for Chapter 9: AREAS OF TRIANGLES
Full solutions for Mathematics for the International Student: Mathematics SL | 3rd Edition
Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!
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).
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
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.
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
Outer product uv T
= column times row = rank one matrix.
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.
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.