######################   VECTORS   ########################
print("VECTORS")

#return DOT PRODUCT of two vectors
def dot_product(u, v):
    if len(u) != len(v):
        print("Error. vectors not same length")
    else:
        return sum([u[i]*v[i] for i in range(len(u))])
        '''
        sum = 0
        for i in range(len(u)):
            sum += u[i] * v[i]
        return sum
        '''

u1=[1,2,3,4,5]
u2=[2,3,4,5,6]
print(dot_product(u1,u2))   #70
u3=[1,0]
u4=[0,1]
u5=[-1,0]
u33=[3,0]
u43=[0,3]
u53=[-3,0]
u6=[.7071,.7071]

print(dot_product(u3,u4))   #0
print(dot_product(u3,u5))   #-1
print(dot_product(u3,u6))   #.7071
print(dot_product(u33,u43))   #0
print(dot_product(u33,u53))   #-9

#return vector multiplied by a scalar
def scalar_multiply(u,a):
    return [a*c for c in u]

print(scalar_multiply(u1,-3))

#return MAGNITUDE (length, norm) of a vector
def magnitude(u):
    return math.sqrt(sum([c**2 for c in u]))
'''
    sum_sqrs = 0
    for c in u:
        sum_sqrs += c**2
    return math.sqrt(sum_sqrs)
'''
print(magnitude(u3))  #1
print(magnitude(u4))  #1
print(magnitude(u5))  #1
print(magnitude(u6))  #1
print(magnitude(u1))  #7.4161

print(math.sqrt(dot_product(u1,u1)))  # root(dot_product of a vector with itself) = norm

#return the ANGLE between two vectors
def angle(u,v):
    return math.acos(dot_product(u,v) / (magnitude(u)*magnitude(v)))

print(angle(u4,u3))  #90 pi/2
print(angle(u5,u3))  #180 pi
print(angle(u53,u33))  #180 pi
print(angle(u43,u33))  #90 pi

#return CROSS PRODUCT of two 3-vectors
def cross_product(u,v):
    if len(u)!=3 or len(v)!=3:
        print("Cross product only of vectors of length 3")
    else:
        uv_cross = []
        uv_cross.append(u[1]*v[2] - u[2]*v[1])
        uv_cross.append(u[2]*v[0] - u[0]*v[2])
        uv_cross.append(u[0]*v[1] - u[1]*v[0])
        return uv_cross

a=[3,-3,1]
b=[4,9,2]
print(cross_product(a,b))  #−15i−2j+39k
#area of parallelogram spanned by two 3-vectors
print(magnitude(cross_product(a,b)))    #41.8330
# = mag(a)*mag(b)*sin alpha
# alpha = arcsin(mag(aXb)/(mag(a)mag(b))
print(math.asin(magnitude(cross_product(a,b))/(magnitude(a)*magnitude(b)))) #1.2694

c=[-12,12,-4]
print(cross_product(a,c))  #0+0+0