{
"metadata": {
"name": "",
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"worksheets": [
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<h1>Chapter 1: Vector Analysis<h1>"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<h3>Example 1.1, Page number: 8<h3>"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
" \n",
"\n",
"import scipy\n",
"from numpy import *\n",
"\n",
"#Variable Declaration\n",
"\n",
"A=array([10,-4,6]) \n",
"B=array([2,1,0])\n",
"ax=array([1,0,0]) #Unit vector along x direction\n",
"ay=array([0,1,0]) #Unit vector along y direction\n",
"az=array([0,0,1]) #Unit vector along z direction\n",
"\n",
"#Calculations\n",
"\n",
"Ay=dot(A,ay) #Component of A along y direction\n",
"l=scipy.sqrt(dot(3*A-B,3*A-B)) #Magnitude of the vector 3A-B\n",
"\n",
" #Defining the x,y and z components of the unit vector along A+2B\n",
" \n",
"ux=round(dot(A+2*B,ax)/scipy.sqrt(dot(A+2*B,A+2*B)),4)\n",
"uy=round(dot(A+2*B,ay)/scipy.sqrt(dot(A+2*B,A+2*B)),4)\n",
"uz=round(dot(A+2*B,az)/scipy.sqrt(dot(A+2*B,A+2*B)),4)\n",
"\n",
"u=array([ux,uy,uz])\n",
"\n",
"#Results\n",
"\n",
"print 'The component of A along y direction is',Ay\n",
"print 'Magnitude of 3A-B =',round(l,2)\n",
"print 'Unit vector along A+2B is',u"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The component of A along y direction is -4\n",
"Magnitude of 3A-B = 35.74\n",
"Unit vector along A+2B is [ 0.9113 -0.1302 0.3906]\n"
]
}
],
"prompt_number": 1
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<h3>Example 1.2, Page number: 9<h3>"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
" \n",
"import scipy\n",
"from numpy import *\n",
"\n",
"#Variable Declaration\n",
"\n",
"P=array([0,2,4]) \n",
"Q=array([-3,1,5])\n",
"ax=array([1,0,0]) #Unit vector along x direction\n",
"ay=array([0,1,0]) #Unit vector along y direction\n",
"az=array([0,0,1]) #Unit vector along z direction\n",
"origin=array([0,0,0]) #Defining the origin\n",
"\n",
"#Calculations\n",
"\n",
"PosP=P-origin #The position vector P\n",
"Dpq=Q-P #The distance vector from P to Q\n",
"l=scipy.sqrt(dot(Dpq,Dpq)) #Magnitude of the distance vector from P to Q\n",
"\n",
" #Defining the x,y and z components of the unit vector along Dpq\n",
" \n",
"ux=round(dot(Dpq,ax)/l,4)\n",
"uy=round(dot(Dpq,ay)/l,4)\n",
"uz=round(dot(Dpq,az)/l,4)\n",
"\n",
"vect=array([ux*10,uy*10,uz*10]) #Vector parallel to PQ with magntude of 10\n",
"\n",
"#Results\n",
"\n",
"print 'The position vector P is',PosP\n",
"print 'The distance vector from P to Q is',Dpq\n",
"print 'The distance between P and Q is',round(l,3)\n",
"print 'Vector parallel to PQ with magntude of 10 is',vect"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The position vector P is [0 2 4]\n",
"The distance vector from P to Q is [-3 -1 1]\n",
"The distance between P and Q is 3.317\n",
"Vector parallel to PQ with magntude of 10 is [-9.045 -3.015 3.015]\n"
]
}
],
"prompt_number": 2
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<h3>Example 1.3, Page number: 10<h3>"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
" \n",
"import scipy\n",
"from numpy import *\n",
"\n",
"#Variable Declaration\n",
"\n",
"vriver=10 #Speed of river in km/hr\n",
"vman=2 #Speed of man relative to boat in km/hr\n",
"ax=array([1,0,0]) #Unit vector along x direction\n",
"ay=array([0,1,0]) #Unit vector along y direction\n",
"az=array([0,0,1]) #Unit vector along z direction\n",
"\n",
"#Calculations\n",
"\n",
" #Velocity of boat\n",
"\n",
"Vboat=vriver*(scipy.cos(scipy.pi/4)*ax- \n",
" scipy.sin(scipy.pi/4)*ay) \n",
" \n",
" #Relative velocity of man with respect to boat\n",
"\n",
"Vrelman=vman*(-scipy.cos(scipy.pi/4)*ax- \n",
" scipy.sin(scipy.pi/4)*ay) \n",
" \n",
" #Absolute velocity of man\n",
" \n",
"Vabs=Vboat+Vrelman\n",
" \n",
"mag=scipy.sqrt(dot(Vabs,Vabs)) #Magnitude of the velocity of man\n",
"Vabsx=dot(Vabs,ax) #X component of absolute velocity\n",
"Vabsy=dot(Vabs,ay) #Y component of absolute velocity\n",
"angle=scipy.arctan(Vabsy/Vabsx)*180/scipy.pi #Angle made with east in degrees\n",
"\n",
"#Result\n",
"\n",
"print 'The velocity of the man is',round(mag,1),'km/hr at an angle of'\n",
"print -round(angle,1),'degrees south of east'\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The velocity of the man is 10.2 km/hr at an angle of\n",
"56.3 degrees south of east\n"
]
}
],
"prompt_number": 3
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<h3>Example 1.4, Page number: 17<h3>"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
" \n",
"\n",
"import scipy\n",
"from numpy import *\n",
"\n",
"#Variable Declaration\n",
"\n",
"A=array([3,4,1])\n",
"B=array([0,2,-5])\n",
"ax=array([1,0,0]) #Unit vector along x direction\n",
"ay=array([0,1,0]) #Unit vector along y direction\n",
"az=array([0,0,1]) #Unit vector along z direction\n",
"\n",
"#Calculations\n",
"\n",
"magA=scipy.sqrt(dot(A,A)) #Magnitude of A\n",
"magB=scipy.sqrt(dot(B,B)) #Magnitude of B\n",
"angle=scipy.arccos(dot(A,B)/(magA*magB)) #Angle between A and B in radians\n",
"angled=angle*180/scipy.pi #Angle between A and B in degrees\n",
"\n",
"\n",
"#Result\n",
"\n",
"print 'The angle between A and B =',round(angled,2),'degrees'\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The angle between A and B = 83.73 degrees\n"
]
}
],
"prompt_number": 4
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<h3>Example 1.5, Page number: 17<h3>"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
" \n",
"\n",
"import scipy\n",
"from numpy import *\n",
"\n",
"#Variable Declaration\n",
"\n",
"P=array([2,0,-1])\n",
"Q=array([2,-1,2])\n",
"R=array([2,-3,1])\n",
"ax=array([1,0,0]) #Unit vector along x direction\n",
"ay=array([0,1,0]) #Unit vector along y direction\n",
"az=array([0,0,1]) #Unit vector along z direction\n",
"\n",
"#Calculations\n",
"\n",
"ansa=cross((P+Q),(P-Q))\n",
"ansb=dot(Q,cross(R,P))\n",
"ansc=dot(P,cross(Q,R))\n",
"lqxr=scipy.sqrt(dot(cross(Q,R),cross(Q,R))) #Magnitude of QXR\n",
"lq=scipy.sqrt(dot(Q,Q)) #Magnitude of Q\n",
"lr=scipy.sqrt(dot(R,R)) #Magnitude of R\n",
"ansd=lqxr/(lq*lr)\n",
"anse=cross(P,cross(Q,R))\n",
"\n",
"#Finding unit vector perpendicular to Q and R\n",
"\n",
"ux=dot(cross(Q,R),ax)/lqxr #X component of the unit vector\n",
"uy=dot(cross(Q,R),ay)/lqxr #Y component of the unit vector\n",
"uz=dot(cross(Q,R),az)/lqxr #Z component of the unit vector\n",
"\n",
"ansf=array([round(ux,3),round(uy,3),round(uz,3)])\n",
"\n",
"ansg=round((float(dot(P,Q))/dot(Q,Q)),4)*Q\n",
"\n",
"\n",
"#Results\n",
"\n",
"print '(P+Q)X(P-Q) =',ansa\n",
"print 'Q.(R X P) =',ansb\n",
"print 'P.(Q X R) =',ansc\n",
"print 'Sin(theta_QR) =',round(ansd,4)\n",
"print 'P X (Q X R) =',anse\n",
"print 'A unit vector perpendicular to both Q and R =',ansf\n",
"print 'The component of P along Q =',ansg\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"(P+Q)X(P-Q) = [ 2 12 4]\n",
"Q.(R X P) = 14\n",
"P.(Q X R) = 14\n",
"Sin(theta_QR) = 0.5976\n",
"P X (Q X R) = [2 3 4]\n",
"A unit vector perpendicular to both Q and R = [ 0.745 0.298 -0.596]\n",
"The component of P along Q = [ 0.4444 -0.2222 0.4444]\n"
]
}
],
"prompt_number": 5
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<h3>Example 1.7, Page number: 21<h3>"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
" \n",
"import scipy\n",
"from numpy import *\n",
"\n",
"#Variable Declaration\n",
"\n",
"P1=array([5,2,-4])\n",
"P2=array([1,1,2])\n",
"P3=array([-3,0,8])\n",
"P4=array([3,-1,0])\n",
"\n",
"#Calculations\n",
"\n",
"R12=P2-P1; #Distance vector from P1 to P2\n",
"R13=P3-P1; #Distance vector from P1 to P3\n",
"R14=P4-P1; #Distance vector from P1 to P4\n",
"s=cross(R12,R13)\n",
"x=cross(R14,R12)\n",
"d=scipy.sqrt(dot(x,x))/scipy.sqrt(dot(R12,R12)) #Distance from line to P4 \n",
"\n",
"#Results\n",
"\n",
"print 'The cross product of the distance vectors R12 and R14 =',s\n",
"print 'So they are along same direction and hence P1, P2 and P3 are collinear.'\n",
"print 'The shortest distance from the line to point P4 =',round(d,3)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The cross product of the distance vectors R12 and R14 = [0 0 0]\n",
"So they are along same direction and hence P1, P2 and P3 are collinear.\n",
"The shortest distance from the line to point P4 = 2.426\n"
]
}
],
"prompt_number": 6
}
],
"metadata": {}
}
]
}