Learn Implementing Vectors in Python | Linear Algebra Foundations

Defining Vectors in Python

In Python, we use NumPy arrays to define 2D vectors like this:


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import numpy as np

v1 = np.array([2, 1])
v2 = np.array([1, 3])

print(f'v1 = {v1}')
print(f'v2 = {v2}')

These represent the vectors:

\vec{v}_1 = (2, 1), \quad \vec{v}_2 = (1, 3)

These can now be added, subtracted, or used in dot product and magnitude calculations.

Vector Addition

To compute vector addition:


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import numpy as np

v1 = np.array([2, 1])
v2 = np.array([1, 3])

v3 = v1 + v2
print(f'v3 = v1 + v2 = {v3}')

This performs:

(2, 1) + (1, 3) = (3, 4)

This matches the rule for vector addition:

\vec{a} + \vec{b} = (a_1 + b_1, \; a_2 + b_2)

Vector Magnitude (Length)

To calculate magnitude in Python:

np.linalg.norm(v)

For vector [3, 4]:


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import numpy as np

print(np.linalg.norm([3, 4]))  # 5.0

This uses the formula:

|\vec{a}| = \sqrt{a_1^2 + a_2^2}

Dot Product

To calculate the dot product:


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import numpy as np

print(np.dot([1, 2], [2, 3]))

Which gives:

[1, 2] \cdot [2, 3] = 1 \cdot 2 + 2 \cdot 3 = 8

Dot product general rule:

\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2

Visualizing Vectors with Matplotlib

You can use the quiver() function in Matplotlib to draw arrows representing vectors and their resultant. Each arrow shows the position, direction, and magnitude of a vector.

Blue: $\vec{v}_1$ , drawn from the origin;
Green: $\vec{v}_2$ , starting at the head of $\vec{v}_1$ ;
Red: resultant vector, drawn from the origin to the final tip.

Example:


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import matplotlib.pyplot as plt

fig, ax = plt.subplots()

# v1
ax.quiver(0, 0, 2, 1, color='blue', angles='xy', scale_units='xy', scale=1)

# v2 (head-to-tail)
ax.quiver(2, 1, 1, 3, color='green', angles='xy', scale_units='xy', scale=1)

# resultant
ax.quiver(0, 0, 3, 4, color='red', angles='xy', scale_units='xy', scale=1)

plt.xlim(0, 5)
plt.ylim(0, 5)
plt.grid(True)
plt.title('Vector Addition (Head-to-Tail Method)')
plt.show()

Parameters (based on the first quiver call):

ax.quiver(0, 0, 2, 1, color='blue', angles='xy', scale_units='xy', scale=1)

0, 0 – starting point of the vector (origin);
2, 1 – vector components in the x and y directions;
color='blue' – sets the arrow color to blue;
angles='xy' – draws the arrow using Cartesian coordinates (x–y plane);
scale_units='xy' – scales the arrow according to the same units as the axes;
scale=1 – keeps the arrow’s true length (no automatic scaling).

This plot shows the head-to-tail vector addition, where the red vector represents the sum $\vec{v}_1 + \vec{v}_2$ .

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Section 4. Chapter 2

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