Vectors and Vector Operations

Nerd Cafe

1. What Is a Vector?

A vector is a mathematical quantity that has:

• Magnitude (length)

• Direction

Example:

Displacement of 5 km North is a vector, but just 5 km is a scalar.

2. Vector Representation

A vector in 2D space is written as:

$$\overrightarrow{V_{1}}=\begin{bmatrix} v_{1} \\ v_{2} \end{bmatrix}$$

import numpy as np

v = np.array([3, 4])
print(v)

Output:

[3 4]

3. Types of Vectors

• Zero vector: [0,0]

• Unit vector: Magnitude is 1

• Equal vectors: Same magnitude and direction

• Opposite vectors: Same magnitude, opposite direction

4. Vector Operations

4.1 Addition & Subtraction

Mathematics:

$$\overrightarrow{A}+\overrightarrow{B}=\begin{bmatrix} a_{1}+b_{1} \\ a_{2}+b_{2} \end{bmatrix}$$

Python:

A = np.array([2, 3])
B = np.array([4, 1])

add = A + B
sub = A - B

print("Addition:", add)
print("Subtraction:", sub)

Output:

Addition: [6 4]
Subtraction: [-2  2]

4.2 Scalar Multiplication

Mathematics:

$$k.\overrightarrow{A}=\begin{bmatrix} k.a_{1} \\ k.a_{2} \end{bmatrix}$$

Python:

v = np.array([2, -1])
k = 3
scaled = k * v

print("Scaled vector:", scaled)

Output:

Scaled vector: [ 6 -3]

4.3 Dot Product

Mathematics:

$$\overrightarrow{A}.\overrightarrow{B}=a_{1}.b_{1}+a_{2}.b_{2}$$

Result is a scalar.

Python:

a = np.array([1, 2])
b = np.array([3, 4])

dot = np.dot(a, b)
print("Dot product:", dot)

Output:

Dot product: 11

Use to find angle or test orthogonality.

4.4 Cross Product (Only in 3D)

Mathematics:

$$\overrightarrow{A}\times \overrightarrow{B}=\begin{bmatrix} a_{2}b_{3}-a_{3}b_{2} \\ a_{3}b_{1}-a_{1}b_{3} \\ a_{1}b_{2}-a_{2}b_{1} \end{bmatrix}$$

Python:

a = np.array([1, 2, 3])
b = np.array([4, 5, 6])

cross = np.cross(a, b)
print("Cross product:", cross)

Output:

Cross product: [-3  6 -3]

4.5 Magnitude or Norm

Mathematics:

$$\left| \overrightarrow{A} \right|=\sqrt{{a_{1}^{2}}+{a_{2}^{2}}}$$

Python:

v = np.array([3, 4])
magnitude = np.linalg.norm(v)

print("Magnitude:", magnitude)

Output:

Magnitude: 5.0

4.6 Unit Vector

Mathematics:

$$\hat{A}=\frac{\overrightarrow{A}}{\left| \overrightarrow{A} \right|}$$

Python:

v = np.array([3, 4])
unit_v = v / np.linalg.norm(v)

print("Unit vector:", unit_v)

Output:

Unit vector: [0.6 0.8]

4.7 Angle Between Vectors

Mathematics:

$$\theta=cos^{-1}(\frac{\overrightarrow{A}.\overrightarrow{B}}{|\overrightarrow{A}||\overrightarrow{B}|})$$

Python:

from numpy import dot, arccos
from numpy.linalg import norm
from numpy import rad2deg

a = np.array([1, 2])
b = np.array([2, 3])

cos_theta = dot(a, b) / (norm(a) * norm(b))
angle_rad = arccos(cos_theta)
angle_deg = rad2deg(angle_rad)

print("Angle (degrees):", angle_deg)

Output:

Angle (degrees): 7.125016348901757

5. Vector Visualization in 2D & 3D

5.1 Plotting a Vector in 2D

Let’s plot a vector from the origin to point (x, y).

Python Code:

import numpy as np
import matplotlib.pyplot as plt

# Define vector
v = np.array([4, 3])
origin = np.array([0, 0])  # Starting point (0,0)

# Plot
plt.figure(figsize=(6, 6))
plt.quiver(*origin, v[0], v[1], angles='xy', scale_units='xy', scale=1, color='blue')
plt.xlim(-1, 6)
plt.ylim(-1, 6)
plt.grid()
plt.gca().set_aspect('equal')
plt.title("2D Vector Plot")
plt.xlabel("X-axis")
plt.ylabel("Y-axis")
plt.show()

plt.quiver() draws an arrow from the origin.

Output:

5.2 Plotting Multiple Vectors in 2D

# Two vectors
a = np.array([2, 3])
b = np.array([4, 1])

# Plot
plt.figure(figsize=(6, 6))
plt.quiver(0, 0, a[0], a[1], color='r', angles='xy', scale_units='xy', scale=1, label='Vector a')
plt.quiver(0, 0, b[0], b[1], color='g', angles='xy', scale_units='xy', scale=1, label='Vector b')
plt.xlim(-1, 6)
plt.ylim(-1, 6)
plt.grid()
plt.gca().set_aspect('equal')
plt.title("Multiple Vectors in 2D")
plt.legend()
plt.show()

Output:

5.3 Plotting Vectors in 3D

from mpl_toolkits.mplot3d import Axes3D

# 3D Vector
v = np.array([2, 3, 4])

# Plot
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.quiver(0, 0, 0, v[0], v[1], v[2], color='purple')

# Axes limits
ax.set_xlim([0, 5])
ax.set_ylim([0, 5])
ax.set_zlim([0, 5])
ax.set_xlabel('X-axis')
ax.set_ylabel('Y-axis')
ax.set_zlabel('Z-axis')
ax.set_title("3D Vector Plot")
plt.show()

6. Plot Vector Addition Graphically

a = np.array([2, 1])
b = np.array([1, 3])
c = a + b  # Resultant

plt.figure(figsize=(6, 6))
plt.quiver(0, 0, a[0], a[1], color='r', scale=1, angles='xy', scale_units='xy', label='a')
plt.quiver(a[0], a[1], b[0], b[1], color='g', scale=1, angles='xy', scale_units='xy', label='b')
plt.quiver(0, 0, c[0], c[1], color='b', scale=1, angles='xy', scale_units='xy', label='a + b')

plt.xlim(-1, 5)
plt.ylim(-1, 5)
plt.grid()
plt.gca().set_aspect('equal')
plt.legend()
plt.title("Vector Addition (Head-to-Tail Method)")
plt.show()

Output: