Higher Engineering Mathematics

242 VECTOR GEOMETRY the sine of the angle between them. The vector prod- uct of vectorsoaandobis written asoa×oband is defined b ...

SCALAR AND VECTOR PRODUCTS 243 D Also, cosθ= a•b ab . Multiplying both sides of this equation bya^2 b^2 and squaring gives: a^2 ...

244 VECTOR GEOMETRY =i(− 12 −84)−j(− 48 −48) +k(− 168 +24) =− 96 i+ 96 j− 144 k or− 48 ( 2 i− 2 j+ 3 k) Practical applications o ...

SCALAR AND VECTOR PRODUCTS 245 D (a) 2p× 3 r(b) (p+r)×q [ (a)− 36 i− 30 j− 54 k (b) 11i+ 4 j−k ] (a)p×(r×q) (b) (3p× 2 r)×q [ ...

246 VECTOR GEOMETRY (b) Find the point on the line corresponding to λ=3 in the resulting equation of part (a). (c) Express the v ...

Assign-06-H8152.tex 19/7/2006 17: 3 Page 247 D Graphs and Vector geometry Assignment 6 This assignment covers the material conta ...

Assign-06-H8152.tex 19/7/2006 17: 3 Page 248 248 GRAPHS AND VECTOR GEOMETRY Determine the work done by a force ofFnewtons actin ...

Complex numbers E 23 Complex numbers 23.1 Cartesian complex numbers (i) If the quadratic equationx^2 + 2 x+ 5 =0is solved using ...

250 COMPLEX NUMBERS Now try the following exercise. Exercise 100 Further problems on the introduction to cartesian complex numbe ...

COMPLEX NUMBERS 251 E −j 4 −j 3 −j j j 2 j 3 Imaginary axis (^034) 5 Real axis R (5−j) P (2+j3) Q (3−j4) 1 2 (a) −j 4 −j 3 −j 2 ...

252 COMPLEX NUMBERS number and its complex conjugate is always a real number. For example, (3+j4)(3−j4)= 9 −j 12 +j 12 −j^216 = ...

COMPLEX NUMBERS 253 E Hence j ( 1 +j 3 1 −j 2 ) 2 =j(−j2)=−j^22 = 2 , sincej^2 =− 1 Now try the following exercise. Exercise 101 ...

254 COMPLEX NUMBERS Multiplying equation (1) by 2 gives: 2 x+ 2 y= 4 (3) Adding equations (2) and (3) gives: −x=7, i.e.,x=− 7 Fr ...

COMPLEX NUMBERS 255 E Modulus, |Z|=r= √ (2^2 + 32 )= √ 13 or 3. 606 , correct to 3 decimal places. Argument,argZ=θ=tan−^132 = 56 ...

256 COMPLEX NUMBERS Hence x=7 cos 35◦= 5. 734 and y=7 sin 35◦= 4. 015 Hence 7∠− 145 ◦=− 5. 734 −j 4. 015 Alternatively 7 ∠− 145 ...

COMPLEX NUMBERS 257 E In Problems 4 and 5 convert the given polar com- plex numbers into (a+jb) form giving answers correct to 4 ...

258 COMPLEX NUMBERS (c) Impedance,Z = 15 ∠− 60 ◦=15[ cos (− 60 ◦)+jsin (− 60 ◦)] = 7. 50 −j 12. 99  Henceresistance= 7. 50 and ...

COMPLEX NUMBERS 259 E CurrentI= V Z =VY =(240∠ 0 ◦)(0. 343 ∠− 15 ◦ 17 ′) = 82. 32 ∠− 15 ◦ 17 ′A Problem 18. Determine the magnit ...

260 COMPLEX NUMBERS 135 ◦to forceA, ForceC, 12 N acting at an angle of 240◦to forceA. [8.394 N, 208◦ 40 ′from forceA] XC = 20 Ω ...

E Complex numbers 24 De Moivre’s theorem 24.1 Introduction From multiplication of complex numbers in polar form, (r∠θ)×(r∠θ)=r^2 ...