H*Problem I .1 .1*LApply@Clear, Names@"Global`*"DD;Off@General::spellD;Off@General::spell1D;vx = -10.;vy = 5.;v = 8vx, vy, 0<;cosa = vx ê Sqrt@vx^ 2 + vy^ 2D;a = ArcCos@cosaD;ad = a 180 ê p;"HaL"Print@" a = arccosHvxê»v»L= ", a, " rad = ", ad, " deg "D;u = v ê Sqrt@vx^2 + vy^2D;Print@" u = vê»v» = ", uD;"HbL"v1 = 5.;q1 = 30.;q1 = q1 p ê 180;v2 = 10.;q2 = 60.;q2 = q2 p ê 180;vv1 = 8v1 Cos@q1D, v1 Sin@q1D, 0<;Print@" v1 = ", vv1D;vv2 = 8v2 Cos@q2D, v2 Sin@q2D, 0<;Print@" v2 = ", vv2D;vv = vv1 + vv2;Print@" v = v1 + v2 = ", vvD;Print@" »v» = ", Sqrt@vv@@1DD^ 2 + vv@@2DD^ 2DD;cosav = vv@@1DD ê Sqrt@vv@@1DD^ 2 + vv@@2DD^ 2D;av = ArcCos@cosavD;adv = av 180 ê p;Print@" av = ", av, " rad = ", adv, " deg "D;HaLa = arccosHvxê»v»L= 2.67795 rad = 153.435 degu = vê»v» = 8-0.894427, 0.447214, 0<HbLv1 = 84.33013, 2.5, 0<v2 = 85., 8.66025, 0<v = v1 + v2 = 89.33013, 11.1603, 0<»v» = 14.5466av = 0.874478 rad = 50.1039 degPI_1_1.nb 1H*Problem I .1 .2*LApply@Clear, Names@"Global`*"DD;Off@General::spellD;Off@General::spell1D;a = P;b = 2 P;c = P Sqrt@2D;a = 45 p ê 180;b = 120 p ê 180;bx = b + p ê 2;g = 30 p ê 180;gx = g + 3 p ê 2;va = 8a Cos@aD, a Sin@aD, 0<;vb = 8b Cos@bxD, b Sin@bxD, 0<;vc = 8c Cos@gxD, c Sin@gxD, 0<;Print@" a = ", vaD;Print@" b = ", vbD;Print@" c = ", vcD;v = Simplify@va + vb + vcD;Print@"v = va+vb+vc = ", vD;vv = Simplify@Sqrt@v@@1DD^ 2 + v@@2DD^ 2DD;Print@"»v» = ", vv, " = ", N@vvDD;cosav = Simplify@v@@1DD ê Sqrt@v@@1DD ^ 2 + v@@2DD^ 2DD;cosav = N@cosav ê. P Ø 1D;av = ArcCos@cosavD;adv = av 180 ê p;Print@" av = ", av, " rad = ", adv, " deg "D;a = 9PÅÅÅÅÅÅÅÅÅÅè!!!!2,PÅÅÅÅÅÅÅÅÅÅè!!!!2, 0=b = 8-è!!!!3 P, -P, 0<c = 9PÅÅÅÅÅÅÅÅÅÅè!!!!2, -$%%%%%%3ÅÅÅÅ2P, 0=v = va+vb+vc = 9Hè!!!!2 -è!!!!3 LP, -1ÅÅÅÅ2H2 -è!!!!2 +è!!!!6 LP, 0=»v» ="#########################################################-H-8 +è!!!!2 +è!!!!3 +è!!!!6 LP2= 1.55056è!!!!!!P2av = 1.77724 rad = 101.828 degPI_1_2.nb 1H*Problem I .1 .3*LApply@Clear, Names@"Global`*"DD;Off@General::spellD;Off@General::spell1D;v1 = 81., 1., 1.<;v2 = 81., 0, 1.<;v3 = 81., 1., 0<;v4 = 80, 1., 1.<;v = v1 + v2 + v3 + v4;Print@"v = l ", vD;cosa = Simplify@v@@1DD ê Sqrt@v@@1DD^ 2 + v@@2DD^ 2 + v@@3DD^ 2DD;a = ArcCos@cosaD;ad = a 180 ê p;Print@" a = ", a, " rad = ", ad, " deg "D;cosb = Simplify@v@@1DD ê Sqrt@v@@1DD^ 2 + v@@2DD^ 2 + v@@3DD^ 2DD;b = ArcCos@cosbD;bd = b 180 ê p;Print@" b = ", b, " rad = ", bd, " deg "D;cosg = Simplify@v@@1DD ê Sqrt@v@@1DD^ 2 + v@@2DD^ 2 + v@@3DD^ 2DD;g = ArcCos@cosgD;gd = b 180 ê p;Print@" g = ", g, " rad = ", gd, " deg "D;v = l 83., 3., 3.<a = 0.955317 rad = 54.7356 degb = 0.955317 rad = 54.7356 degg = 0.955317 rad = 54.7356 degPI_1_3.nb 1H*Problem I .1 .4*LApply@Clear, Names@"Global`*"DD;Off@General::spellD;Off@General::spell1D;v1 = 8-3., 4., -3.<;v2 = 83., 0, 3.<;v3 = 81., 2., 3<;E1 = v1 + v2 + v3;Print@"E1 = v1+v2+v3 = ", E1D;E2 = v1 + v2 - v3;Print@"E2 = v1+v2-v3 = ", E2D;E3 = Cross@Cross@v1 , v2 D, v3D;Print@"E3 = Hv1 x v2L x v3 = ", E3D;E4 = Cross@v1 , v2 D.v3;Print@"E4 = Hv1 x v2L . v3 = ", E4D;E1 = v1+v2+v3 = 81., 6., 3.<E2 = v1+v2-v3 = 8-1., 2., -3.<E3 = Hv1 x v2L x v3 = 824., -48., 24.<E4 = Hv1 x v2L . v3 = -24.PI_1_4.nb 1H*Problem I .1 .5*LApply@Clear, Names@"Global`*"DD;Off@General::spellD;Off@General::spell1D;v1 = 82., -4., 4.<;v2 = 84., 2, 4.<;vd = v1 .v2 ;Print@"v1 . v2 = ", vdD;vc = Cross@v1 , v2 D;Print@"v1 x v2 = ", vcD;vm = [email protected];"»v1 x v2»=»v1» »v2» sina =>""sina=»v1 x v2»êH»v1» »v2»L"sina = vm ê [email protected] [email protected];a = ArcSin@sinaD;ad = a 180 ê p;Print@" a = ", a, " rad = ", ad, " deg "D;"v1 . v2»=»v1» »v2» cosa =>""cosa=Hv1 . v2LêH»v1» »v2»L"cosad = vd ê [email protected] [email protected];ad = ArcCos@cosadD;add = a 180 ê p;Print@" a = ", ad, " rad = ", add, " deg "D;v1 . v2 = 16.v1 x v2 = 8-24., 8., 20.<»v1 x v2»=»v1» »v2» sina =>sina=»v1 x v2»êH»v1» »v2»La = 1.11024 rad = 63.6122 degv1 . v2»=»v1» »v2» cosa =>cosa=Hv1 . v2LêH»v1» »v2»La = 1.11024 rad = 63.6122 degPI_1_5.nb 1H*Problem I .1 .6*LApply@Clear, Names@"Global`*"DD;Off@General::spellD;Off@General::spell1D;v1 = 82., 4., 6.<;v2 = 81., 3, 5.<;v3 = 8-2., 0., 2<;"v1 x H v2 x v3L=v1.v3 v2 - v1.v2 v3";v = Cross@v1, Cross@v2, v3DD;Print@"v1 x H v2 x v3L = ", vD;vc = v1.v3 v2 - v1.v2 v3;Print@"v1.v3 v2-v1.v2 v3 = ", vcD;vm = v1.Cross@v2, v3D;Print@"v1 . H v2 x v3L = ", vmD;"v1 . H v2 x v3L = 0 => the vectors are in the sameplane and the vector v3 can be written as a linear combinationof the vectors v1 and v3: v3= l v1 + m v2, l  0, m  0"eq = v3 - l v1 - m v2;Print@"v3-l v1 - m v2 = 0 =>", eq, " = 0"De1 = eq@@1DD ã 0;e2 = eq@@2DD ã 0;e3 = eq@@3DD ã 0;"Projected on x,y,z give"Print@eq@@1DD, " = 0"DPrint@eq@@2DD, " = 0"DPrint@eq@@3DD, " = 0"Dss = Solve@8e1, e3<, 8l, m<D;ls = l ê. ss@@1DD;ms = m ê. ss@@1DD;Print@" l = ", lsD;Print@" m = ", msD;v1 x H v2 x v3L = 896., 24., -48.<v1.v3 v2-v1.v2 v3 = 896., 24., -48.<v1 . H v2 x v3L = 0.v1 . H v2 x v3L = 0 => the vectors are in the sameplane and the vector v3 can be written as a linear combinationof the vectors v1 and v3: v3= l v1 + m v2, l  0, m  0v3-l v1 - m v2 = 0 =>8-2. - 2. l - 1. m, 0. - 4. l - 3 m, 2 - 6. l - 5. m< = 0Projected on x,y,z give-2. - 2. l - 1. m = 00. - 4. l - 3 m = 02 - 6. l - 5. m = 0l = -3.m = 4.PI_1_6.nb 1PI.1.7. Solve the vectorial equation x × a = x × b where a and b are twoknown vectors.Solutionx × a = x × b <=> x × (a − b) = 0 =⇒x=0 or x collinear with the difference a − b.1PI.1.8. Solve the vec torial equation v = a × x where v and a are twoknown vectors.Solutionv = a × x =⇒ v⊥plane(a, x) =⇒ v⊥xv = |v| = a × x| =⇒ v = a x sin α.x sin α = OA is the projection of x on the direction perpendicular to a soOA = v/a = constant.It results that the extremity (end) of the vector x can move on a lineparallel to a at a distance v/a from the vector a.1Figure PI.1.8vaxxxαa=AOPI.1.9. Solve the vectorial equation a · x = m, where a is a known givenvector and m is a known given scalar.Solutiona · x = x a cos α = a OA where x cos α = OA.a OA = m =⇒ OA = m/a.The solution of the vectorial equation a · x = m is an undetermined vectorx = rOMwith the same origin …