In [1]:

```
import scipy
from numpy import *
#Variable Declaration
A=array([10,-4,6])
B=array([2,1,0])
ax=array([1,0,0]) #Unit vector along x direction
ay=array([0,1,0]) #Unit vector along y direction
az=array([0,0,1]) #Unit vector along z direction
#Calculations
Ay=dot(A,ay) #Component of A along y direction
l=scipy.sqrt(dot(3*A-B,3*A-B)) #Magnitude of the vector 3A-B
#Defining the x,y and z components of the unit vector along A+2B
ux=round(dot(A+2*B,ax)/scipy.sqrt(dot(A+2*B,A+2*B)),4)
uy=round(dot(A+2*B,ay)/scipy.sqrt(dot(A+2*B,A+2*B)),4)
uz=round(dot(A+2*B,az)/scipy.sqrt(dot(A+2*B,A+2*B)),4)
u=array([ux,uy,uz])
#Results
print 'The component of A along y direction is',Ay
print 'Magnitude of 3A-B =',round(l,2)
print 'Unit vector along A+2B is',u
```

In [2]:

```
import scipy
from numpy import *
#Variable Declaration
P=array([0,2,4])
Q=array([-3,1,5])
ax=array([1,0,0]) #Unit vector along x direction
ay=array([0,1,0]) #Unit vector along y direction
az=array([0,0,1]) #Unit vector along z direction
origin=array([0,0,0]) #Defining the origin
#Calculations
PosP=P-origin #The position vector P
Dpq=Q-P #The distance vector from P to Q
l=scipy.sqrt(dot(Dpq,Dpq)) #Magnitude of the distance vector from P to Q
#Defining the x,y and z components of the unit vector along Dpq
ux=round(dot(Dpq,ax)/l,4)
uy=round(dot(Dpq,ay)/l,4)
uz=round(dot(Dpq,az)/l,4)
vect=array([ux*10,uy*10,uz*10]) #Vector parallel to PQ with magntude of 10
#Results
print 'The position vector P is',PosP
print 'The distance vector from P to Q is',Dpq
print 'The distance between P and Q is',round(l,3)
print 'Vector parallel to PQ with magntude of 10 is',vect
```

In [3]:

```
import scipy
from numpy import *
#Variable Declaration
vriver=10 #Speed of river in km/hr
vman=2 #Speed of man relative to boat in km/hr
ax=array([1,0,0]) #Unit vector along x direction
ay=array([0,1,0]) #Unit vector along y direction
az=array([0,0,1]) #Unit vector along z direction
#Calculations
#Velocity of boat
Vboat=vriver*(scipy.cos(scipy.pi/4)*ax-
scipy.sin(scipy.pi/4)*ay)
#Relative velocity of man with respect to boat
Vrelman=vman*(-scipy.cos(scipy.pi/4)*ax-
scipy.sin(scipy.pi/4)*ay)
#Absolute velocity of man
Vabs=Vboat+Vrelman
mag=scipy.sqrt(dot(Vabs,Vabs)) #Magnitude of the velocity of man
Vabsx=dot(Vabs,ax) #X component of absolute velocity
Vabsy=dot(Vabs,ay) #Y component of absolute velocity
angle=scipy.arctan(Vabsy/Vabsx)*180/scipy.pi #Angle made with east in degrees
#Result
print 'The velocity of the man is',round(mag,1),'km/hr at an angle of'
print -round(angle,1),'degrees south of east'
```

In [4]:

```
import scipy
from numpy import *
#Variable Declaration
A=array([3,4,1])
B=array([0,2,-5])
ax=array([1,0,0]) #Unit vector along x direction
ay=array([0,1,0]) #Unit vector along y direction
az=array([0,0,1]) #Unit vector along z direction
#Calculations
magA=scipy.sqrt(dot(A,A)) #Magnitude of A
magB=scipy.sqrt(dot(B,B)) #Magnitude of B
angle=scipy.arccos(dot(A,B)/(magA*magB)) #Angle between A and B in radians
angled=angle*180/scipy.pi #Angle between A and B in degrees
#Result
print 'The angle between A and B =',round(angled,2),'degrees'
```

In [5]:

```
import scipy
from numpy import *
#Variable Declaration
P=array([2,0,-1])
Q=array([2,-1,2])
R=array([2,-3,1])
ax=array([1,0,0]) #Unit vector along x direction
ay=array([0,1,0]) #Unit vector along y direction
az=array([0,0,1]) #Unit vector along z direction
#Calculations
ansa=cross((P+Q),(P-Q))
ansb=dot(Q,cross(R,P))
ansc=dot(P,cross(Q,R))
lqxr=scipy.sqrt(dot(cross(Q,R),cross(Q,R))) #Magnitude of QXR
lq=scipy.sqrt(dot(Q,Q)) #Magnitude of Q
lr=scipy.sqrt(dot(R,R)) #Magnitude of R
ansd=lqxr/(lq*lr)
anse=cross(P,cross(Q,R))
#Finding unit vector perpendicular to Q and R
ux=dot(cross(Q,R),ax)/lqxr #X component of the unit vector
uy=dot(cross(Q,R),ay)/lqxr #Y component of the unit vector
uz=dot(cross(Q,R),az)/lqxr #Z component of the unit vector
ansf=array([round(ux,3),round(uy,3),round(uz,3)])
ansg=round((float(dot(P,Q))/dot(Q,Q)),4)*Q
#Results
print '(P+Q)X(P-Q) =',ansa
print 'Q.(R X P) =',ansb
print 'P.(Q X R) =',ansc
print 'Sin(theta_QR) =',round(ansd,4)
print 'P X (Q X R) =',anse
print 'A unit vector perpendicular to both Q and R =',ansf
print 'The component of P along Q =',ansg
```

In [6]:

```
import scipy
from numpy import *
#Variable Declaration
P1=array([5,2,-4])
P2=array([1,1,2])
P3=array([-3,0,8])
P4=array([3,-1,0])
#Calculations
R12=P2-P1; #Distance vector from P1 to P2
R13=P3-P1; #Distance vector from P1 to P3
R14=P4-P1; #Distance vector from P1 to P4
s=cross(R12,R13)
x=cross(R14,R12)
d=scipy.sqrt(dot(x,x))/scipy.sqrt(dot(R12,R12)) #Distance from line to P4
#Results
print 'The cross product of the distance vectors R12 and R14 =',s
print 'So they are along same direction and hence P1, P2 and P3 are collinear.'
print 'The shortest distance from the line to point P4 =',round(d,3)
```