Vector similarity metrics in Prompt Engineering / GenAI - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to calculate the cosine similarity between two vectors.

Prompt Engineering / GenAI

from numpy import dot
from numpy.linalg import norm

def cosine_similarity(vec1, vec2):
    return dot(vec1, vec2) / (norm(vec1) [1] norm(vec2))

Drag options to blanks, or click blank then click option'

A*

B+

C-

D/

Attempts:

3 left

💡 Hint

Common Mistakes

Using addition instead of multiplication between the norms.

Dividing one norm by the other instead of multiplying.

The cosine similarity formula divides the dot product by the product of the norms (lengths) of the two vectors, so the operator between the norms is multiplication.

2fill in blank

medium

Complete the code to compute the Euclidean distance between two vectors.

Prompt Engineering / GenAI

import numpy as np

def euclidean_distance(vec1, vec2):
    return np.sqrt(np.sum((vec1 - vec2) [1] 2))

Drag options to blanks, or click blank then click option'

A+

B**

C*

D//

Attempts:

3 left

💡 Hint

Common Mistakes

Using multiplication '*' instead of exponent '**'.

Using integer division '//' which causes errors.

Euclidean distance uses the square of the difference, so the exponent operator '**' with 2 is correct.

3fill in blank

hard

Fix the error in the code to calculate the Manhattan distance between two vectors.

Prompt Engineering / GenAI

import numpy as np

def manhattan_distance(vec1, vec2):
    return np.sum(np.abs(vec1 [1] vec2))

Drag options to blanks, or click blank then click option'

A/

B+

C*

D-

Attempts:

3 left

💡 Hint

Common Mistakes

Using addition instead of subtraction inside the absolute value.

Using multiplication or division which are incorrect here.

Manhattan distance sums the absolute differences between vector elements, so subtraction is needed inside abs().

4fill in blank

hard

Fill both blanks to create a dictionary of word vectors filtered by length greater than 3.

Prompt Engineering / GenAI

words = ['cat', 'house', 'tree', 'sun']
vectors = {'cat': [1,2], 'house': [3,4], 'tree': [5,6], 'sun': [7,8]}
filtered_vectors = {word: vectors[word] for word in words if [1] [2] 3}

Drag options to blanks, or click blank then click option'

Alen(word)

Bword

C>

D<

Attempts:

3 left

💡 Hint

Common Mistakes

Using the word itself instead of its length.

Using less than '<' instead of greater than '>'.

The condition filters words whose length is greater than 3, so 'len(word) > 3' is correct.

5fill in blank

hard

Fill all three blanks to create a dictionary of words and their vector lengths filtered by length greater than 3.

Prompt Engineering / GenAI

words = ['dog', 'elephant', 'bird', 'ant']
vectors = {'dog': [1,1,1], 'elephant': [2,2,2], 'bird': [3,3,3], 'ant': [4,4,4]}
result = [1]: [2] for word in words if len(word) [3] 3

Drag options to blanks, or click blank then click option'

Aword

Blen(vectors[word])

C>

Dword.upper()

Attempts:

3 left

💡 Hint

Common Mistakes

Using the word as key without uppercase.

Using the word length instead of vector length as value.

Using '<' instead of '>' in the condition.

The dictionary keys are uppercase words, values are vector lengths, filtered by word length greater than 3.

Practice

1. Which vector similarity metric measures the angle between two vectors to determine how similar they are?

easy

A. Manhattan distance

B. Euclidean distance

C. Cosine similarity

D. Jaccard similarity

Solution

1. Step 1: Understand cosine similarity

   Cosine similarity measures the cosine of the angle between two vectors, showing how aligned they are.

2. Step 2: Compare with other metrics

   Euclidean and Manhattan distances measure gaps, not angles. Jaccard is for sets, not vectors.

3. Final Answer:

   Cosine similarity -> Option C

4. Quick Check:

   Angle-based similarity = Cosine similarity [OK]

Hint: Angle means cosine similarity, distance means Euclidean [OK]

Common Mistakes:

• Confusing distance with angle measurement
• Thinking Euclidean measures angle
• Mixing set similarity with vector similarity

2. Which of the following is the correct Python expression to compute cosine similarity between two vectors a and b using numpy?

easy

A. np.linalg.norm(a - b)

B. np.dot(a, b) * (np.linalg.norm(a) + np.linalg.norm(b))

C. np.sum(np.abs(a - b))

D. np.dot(a, b) / (np.linalg.norm(a) * np.linalg.norm(b))

Solution

1. Step 1: Recall cosine similarity formula

   Cosine similarity = dot product of vectors divided by product of their lengths (norms).

2. Step 2: Match formula to code

   np.dot(a, b) / (np.linalg.norm(a) * np.linalg.norm(b)) matches this formula exactly. np.linalg.norm(a - b) is Euclidean distance, C is Manhattan distance, D is incorrect formula.

3. Final Answer:

   np.dot(a, b) / (np.linalg.norm(a) * np.linalg.norm(b)) -> Option D

4. Quick Check:

   Dot product over norms = cosine similarity [OK]

Hint: Cosine = dot product divided by norms product [OK]

Common Mistakes:

• Using subtraction instead of dot product
• Multiplying norms instead of dividing
• Confusing Euclidean with cosine formula

3. Given vectors a = np.array([1, 2, 3]) and b = np.array([4, 5, 6]), what is the output of np.linalg.norm(a - b)?

medium

A. 3.742

B. 5.196

C. 15.0

D. 32.0

Solution

1. Step 1: Calculate vector difference

   a - b = [1-4, 2-5, 3-6] = [-3, -3, -3]

2. Step 2: Compute Euclidean norm

   Norm = sqrt((-3)^2 + (-3)^2 + (-3)^2) = sqrt(9+9+9) = sqrt(27) ≈ 5.196

3. Final Answer:

   5.196 -> Option B

4. Quick Check:

   Euclidean distance = 5.196 [OK]

Hint: Euclidean norm = sqrt(sum of squared differences) [OK]

Common Mistakes:

• Forgetting to square differences
• Calculating sum instead of sqrt of sum
• Mixing up vector subtraction order

4. Identify the error in this Python code snippet for cosine similarity:

import numpy as np

def cosine_sim(a, b):
    return np.dot(a, b) / np.linalg.norm(a) + np.linalg.norm(b)

print(cosine_sim(np.array([1, 0]), np.array([0, 1])))

medium

A. The denominator should multiply norms, not add them

B. np.dot is used incorrectly; should be np.cross

C. Vectors must be normalized before dot product

D. Function is missing return statement

Solution

1. Step 1: Analyze denominator in formula

   The code adds norms: np.linalg.norm(a) + np.linalg.norm(b), but cosine similarity divides by their product.

2. Step 2: Understand correct formula

   Cosine similarity = dot(a,b) / (norm(a) * norm(b)), so addition is wrong here.

3. Final Answer:

   The denominator should multiply norms, not add them -> Option A

4. Quick Check:

   Denominator = product of norms [OK]

Hint: Denominator in cosine similarity multiplies norms [OK]

Common Mistakes:

• Adding norms instead of multiplying
• Using cross product instead of dot product
• Forgetting to return value

5. You have two text documents represented as vectors: doc1 = [1, 0, 2, 1] and doc2 = [0, 1, 1, 1]. Which similarity metric is best to find how similar their topics are, and why?

hard

A. Cosine similarity, because it measures angle ignoring length differences

B. Euclidean distance, because it measures exact gap between vectors

C. Manhattan distance, because it sums absolute differences

D. Jaccard similarity, because it compares set overlap

Solution

1. Step 1: Understand vector meaning in text

   Vectors represent word counts or weights; length can vary by document size.

2. Step 2: Choose metric ignoring length but capturing direction

   Cosine similarity measures angle, so it focuses on topic similarity ignoring document length differences.

3. Final Answer:

   Cosine similarity, because it measures angle ignoring length differences -> Option A

4. Quick Check:

   Topic similarity = cosine similarity [OK]

Hint: For text, angle-based similarity works best [OK]

Common Mistakes:

• Using Euclidean which is sensitive to length
• Confusing set similarity with vector similarity
• Ignoring document length effect