Vectors are used to represent direction, magnitude, and position in many fields — including data science, where they model feature sets, weights, embeddings, and more.

Defining Vectors in Python

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


              1234567
            
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:


              1234567
            
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]:


              123
            
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:


              123
            
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

We can use quiver() to draw arrows representing vectors. Here's what each one does:

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 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')

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

# resultant
ax.quiver(0, 0, 3, 4, color='red')

plt.xlim(0, 5)
plt.ylim(0, 5)
plt.grid(True)
plt.show()

This produces a triangle that visualizes vector addition.

Quiz

Correct Answer: B

Correct Answer: C

Because:

\sqrt{3^2 + 4^2} = 5

Correct Answer: B

Correct Answer: C

Tudo estava claro?

Obrigado pelo seu feedback!

Seção 4. Capítulo 2

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Vectors are used to represent direction, magnitude, and position in many fields — including data science, where they model feature sets, weights, embeddings, and more.

Defining Vectors in Python

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


              1234567
            
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:


              1234567
            
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]:


              123
            
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:


              123
            
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

We can use quiver() to draw arrows representing vectors. Here's what each one does:

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 origin to the final tip.

Example:


              1234567891011121314151617
            
import matplotlib.pyplot as plt

fig, ax = plt.subplots()

# v1
ax.quiver(0, 0, 2, 1, color='blue')

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

# resultant
ax.quiver(0, 0, 3, 4, color='red')

plt.xlim(0, 5)
plt.ylim(0, 5)
plt.grid(True)
plt.show()

This produces a triangle that visualizes vector addition.

Quiz

Correct Answer: B

Correct Answer: C

Because:

\sqrt{3^2 + 4^2} = 5

Correct Answer: B

Correct Answer: C

Tudo estava claro?

Obrigado pelo seu feedback!

Seção 4. Capítulo 2

Implementing Vectors in Python

Defining Vectors in Python

Vector Addition

Vector Magnitude (Length)

Dot Product

Visualizing Vectors with Matplotlib

Quiz

1. What is the result of:

2. What is the magnitude of:

3. Which code correctly computes the dot product of $[1,2]$ and $[2,3]$ ?

4. What does the resultant vector represent in head-to-tail addition?

5. How can you display the magnitude of the resultant vector on the plot?

Awesome!

Implementing Vectors in Python

Defining Vectors in Python

Vector Addition

Vector Magnitude (Length)

Dot Product

Visualizing Vectors with Matplotlib

Quiz

1. What is the result of:

2. What is the magnitude of:

3. Which code correctly computes the dot product of $[1,2]$ and $[2,3]$ ?

4. What does the resultant vector represent in head-to-tail addition?

5. How can you display the magnitude of the resultant vector on the plot?

Implementing Vectors in Python

Defining Vectors in Python

Vector Addition

Vector Magnitude (Length)

Dot Product

Visualizing Vectors with Matplotlib

Quiz

1. What is the result of:

2. What is the magnitude of:

3. Which code correctly computes the dot product of [1,2][1,2][1,2] and [2,3][2,3][2,3]?

4. What does the resultant vector represent in head-to-tail addition?

5. How can you display the magnitude of the resultant vector on the plot?

Awesome!

Implementing Vectors in Python

Defining Vectors in Python

Vector Addition

Vector Magnitude (Length)

Dot Product

Visualizing Vectors with Matplotlib

Quiz

1. What is the result of:

2. What is the magnitude of:

3. Which code correctly computes the dot product of [1,2][1,2][1,2] and [2,3][2,3][2,3]?

4. What does the resultant vector represent in head-to-tail addition?

5. How can you display the magnitude of the resultant vector on the plot?

3. Which code correctly computes the dot product of $[1,2]$ and $[2,3]$ ?

3. Which code correctly computes the dot product of $[1,2]$ and $[2,3]$ ?