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Mathematics for Data Science

bookImplementing Vectors in Python

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}')
copy

These represent the vectors:

v⃗1=(2,1),v⃗2=(1,3)\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}')
copy

This performs:

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

This matches the rule for vector addition:

a⃗+b⃗=(a1+b1,  a2+b2)\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
copy

This uses the formula:

∣a⃗∣=a12+a22|\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]))
copy

Which gives:

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

Dot product general rule:

a⃗⋅b⃗=a1b1+a2b2\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: v⃗1\vec{v}_1, drawn from the origin;
  • Green: v⃗2\vec{v}_2, starting at the head of v⃗1\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()
copy

This produces a triangle that visualizes vector addition.

Quiz

Correct Answer: B

Correct Answer: C

Because:

32+42=5\sqrt{3^2 + 4^2} = 5

Correct Answer: B

Correct Answer: C

Correct Answer: C

1. What is the result of:

[2,1]+[1,3][2, 1] + [1, 3]

2. What is the magnitude of:

(3,4)(3, 4)

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

question mark

What is the result of:

[2,1]+[1,3][2, 1] + [1, 3]

Select the correct answer

question mark

What is the magnitude of:

(3,4)(3, 4)

Select the correct answer

question mark

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

Select the correct answer

question mark

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

Select the correct answer

question mark

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

Select the correct answer

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Як ми можемо покращити це?

Дякуємо за ваш відгук!

Секція 4. Розділ 2

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bookImplementing Vectors in Python

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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}')
copy

These represent the vectors:

v⃗1=(2,1),v⃗2=(1,3)\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}')
copy

This performs:

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

This matches the rule for vector addition:

a⃗+b⃗=(a1+b1,  a2+b2)\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
copy

This uses the formula:

∣a⃗∣=a12+a22|\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]))
copy

Which gives:

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

Dot product general rule:

a⃗⋅b⃗=a1b1+a2b2\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: v⃗1\vec{v}_1, drawn from the origin;
  • Green: v⃗2\vec{v}_2, starting at the head of v⃗1\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()
copy

This produces a triangle that visualizes vector addition.

Quiz

Correct Answer: B

Correct Answer: C

Because:

32+42=5\sqrt{3^2 + 4^2} = 5

Correct Answer: B

Correct Answer: C

Correct Answer: C

1. What is the result of:

[2,1]+[1,3][2, 1] + [1, 3]

2. What is the magnitude of:

(3,4)(3, 4)

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

question mark

What is the result of:

[2,1]+[1,3][2, 1] + [1, 3]

Select the correct answer

question mark

What is the magnitude of:

(3,4)(3, 4)

Select the correct answer

question mark

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

Select the correct answer

question mark

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

Select the correct answer

question mark

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

Select the correct answer

Все було зрозуміло?

Як ми можемо покращити це?

Дякуємо за ваш відгук!

Секція 4. Розділ 2
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