Vectors
Contents
## This cell just imports necessary modules
import numpy as np
import plotly.graph_objs as go
# This cell shows a function that can plot vectors using the Scatter3d() in plotly.
#In the plots the vectors have a big point to mark the direction.
def vector_plot(tvects,is_vect=True,orig=[0,0,0]):
"""Plot vectors using plotly"""
if is_vect:
if not hasattr(orig[0],"__iter__"):
coords = [[orig,np.sum([orig,v],axis=0)] for v in tvects]
else:
coords = [[o,np.sum([o,v],axis=0)] for o,v in zip(orig,tvects)]
else:
coords = tvects
data = []
for i,c in enumerate(coords):
X1, Y1, Z1 = zip(c[0])
X2, Y2, Z2 = zip(c[1])
vector = go.Scatter3d(x = [X1[0],X2[0]],
y = [Y1[0],Y2[0]],
z = [Z1[0],Z2[0]],
marker = dict(size = [0,5],
color = ['blue'],
line=dict(width=5,
color='DarkSlateGrey')),
name = 'Vector'+str(i+1))
data.append(vector)
layout = go.Layout(
margin = dict(l = 4,
r = 4,
b = 4,
t = 4)
)
fig = go.Figure(data=data,layout=layout)
fig.show()
Vectors#
This notebook will illustrate how to apply the maths discussed in the lecture using Python.
In these notebooks, we will adopt the following prefix convention when naming variables:
's' (e.g. sDotProduct) means the variable is a scalar
'v' (e.g. vCrossProduct) means the variable is a vector
'm' (e.g. mA) means the variable is a matrix
# Let's define two vectors, by listing their components
vA = [1, 2, 1]
vB = [-1, 1, 0]
# Convert vA and vB from a list to an array so we can use numpy
# to perform vector operations on them.
vA = np.array(vA)
vB = np.array(vB)
print("Plot of vA (blue) and vB (red)")
vector_plot([vA,vB])
Plot of vA (blue) and vB (red)
Null vector#
Slide 9
# vNull is the null vector (a vector of all zeros,
# created using the zeros function)
vNull = np.zeros(3)
print("The null vector is: ", vNull)
The null vector is: [0. 0. 0.]
Vector equality#
Slide 9
# Here we compare vectors vA and vB. This is done element-by-element.
# The .all() function allows us to check that the elements of vA.
# are ALL equal to those of vB. In other words, vA[i] == vB[i] for i = 1, 2, 3
if(np.equal(vA,vB).all()):
print("Vectors vA and vB are equal")
else:
print("Vectors vA and vB are NOT equal")
Vectors vA and vB are NOT equal
Multiplication of a vector by a scalar#
Slide 9
print("2*vA = ", 2*vA)
print("10*vB = ", 10*vB)
print("Plot of 2*vA (blue) and 10*vB (red)")
vector_plot([2*vA,10*vB])
2*vA = [2 4 2]
10*vB = [-10 10 0]
Plot of 2*vA (blue) and 10*vB (red)
Vector addition and subtraction#
Slide 10
print("vA + vB = ", vA + vB)
print("vA - vB = ", vA - vB)
print("Plot of vA + vB (blue) and vA - vB (red)")
vector_plot([vA + vB,vA - vB])
vA + vB = [0 3 1]
vA - vB = [2 1 1]
Plot of vA + vB (blue) and vA - vB (red)
Vector magnitude#
Slide 7
Compute the magnitude (i.e. Euclidean/L2 norm) of vectors vA and vB (to 3 significant figures). There are many ways to do this in Python. Try using the sqrt and dot functions (see below) to achieve the same result.
sMagnitude_vA = np.linalg.norm(vA, ord=2)
sMagnitude_vB = np.linalg.norm(vB, ord=2)
print("Magnitude of vector vA is %.3f" % sMagnitude_vA)
print("Magnitude of vector vB is %.3f" % sMagnitude_vB)
Magnitude of vector vA is 2.449
Magnitude of vector vB is 1.414
Dot product#
Slide 15
Remember: the dot product of two vectors always returns a scalar
sDotProduct_vAvB = np.dot(vA,vB)
print("The dot product of vA and vB is %.3f" % sDotProduct_vAvB)
The dot product of vA and vB is 1.000
Angle between two vectors#
Slide 18
# NOTE: arccos is inverse cos
sTheta = np.arccos(sDotProduct_vAvB/(sMagnitude_vA*sMagnitude_vB))
print("The angle between vA and vB is %.3f radians (or %.3f degrees)"
% (sTheta, sTheta*(180/np.pi)))
The angle between vA and vB is 1.278 radians (or 73.221 degrees)
Cross product#
Slide 19
Remember: the cross product of two vectors always returns another vector. We could use np.cross:
print("The cross product of vA and vB is", np.cross(vA,vB))
The cross product of vA and vB is [-1 -1 3]
Alternatively, we could compute determinants as in slide 22:
# Remember: in Python, array indices always start from zero,
# so the x, y and z components have an index of 0, 1 and 2 respectively.
vCrossProduct = [ np.linalg.det([[vA[1], vA[2]],
[vB[1], vB[2]]]),
np.linalg.det([[vA[0], vA[2]],
[vB[0], vB[2]]]),
np.linalg.det([[vA[0], vA[1]],
[vB[0], vB[1]]]) ]
print("The cross product of vA and vB is", vCrossProduct)
print("Plot of vA (blue),vB (red) and the cross product of vA and vB (green)")
vector_plot([vA,vB,np.cross(vA,vB)])
The cross product of vA and vB is [-1.0, 1.0, 3.0000000000000004]
Plot of vA (blue),vB (red) and the cross product of vA and vB (green)