Vector similarity metrics in Prompt Engineering / GenAI - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the purpose of vector similarity metrics in machine learning?

Vector similarity metrics measure how alike two vectors are. They help compare data points, like checking if two images or texts are similar.

beginner

Explain Cosine Similarity in simple terms.

Cosine Similarity measures the angle between two vectors. If the angle is small, the vectors are similar. It ignores their length and focuses on direction.

beginner

What is Euclidean Distance and how does it relate to similarity?

Euclidean Distance is the straight-line distance between two points (vectors). Smaller distance means higher similarity, bigger distance means less similar.

intermediate

How does Jaccard Similarity work for comparing vectors?

Jaccard Similarity compares two sets by dividing the size of their overlap by the size of their union. For vectors, it measures how many features they share compared to total features.

intermediate

Why might you choose Cosine Similarity over Euclidean Distance?

Cosine Similarity focuses on direction, ignoring length, which is useful when magnitude varies but pattern matters. Euclidean Distance considers magnitude, which can be misleading if scale differs.

Which metric measures the angle between two vectors?

A Cosine Similarity

B Euclidean Distance

C Jaccard Similarity

D Manhattan Distance

Cosine Similarity calculates the cosine of the angle between two vectors to measure similarity.

If two vectors have a Euclidean distance of zero, what does that mean?

A They are identical

B They are completely different

C They have no features

D They have opposite directions

A Euclidean distance of zero means the vectors are exactly the same point in space.

Jaccard Similarity is best used for comparing:

A Angles between vectors

B Continuous numeric vectors

C Distances in 3D space

D Sets or binary vectors

Jaccard Similarity compares overlap between sets or binary vectors.

Which similarity metric ignores the length of vectors and focuses on direction?

A Euclidean Distance

B Hamming Distance

C Cosine Similarity

D Jaccard Similarity

Cosine Similarity measures the angle between vectors, ignoring their length.

A higher Euclidean distance between two vectors means:

A They are more similar

B They are less similar

C They have the same direction

D They have identical features

Greater Euclidean distance means vectors are farther apart and less similar.

Describe three common vector similarity metrics and when you might use each.

Think about what each metric focuses on: angle, distance, or overlap.

Explain why choosing the right similarity metric matters in machine learning tasks.

Consider how data features and scale affect similarity.

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