Vector Angular Similarity Measure

In vector mathematics and machine learning, measuring the directional similarity between two vectors is a fundamental operation that captures how similarly oriented they are in space, independent of their magnitudes. This concept, known as angular similarity, quantifies the relationship between vectors based purely on the angle between them.

The angular similarity between two vectors v₁ and v₂ is computed by taking the normalized dot product of the vectors—dividing their dot product by the product of their magnitudes (Euclidean norms):

$$\text{angular_similarity}(v_1, v_2) = \frac{v_1 \cdot v_2}{|v_1| \times |v_2|} = \frac{\sum_{i=1}^{n} v_{1i} \cdot v_{2i}}{\sqrt{\sum_{i=1}^{n} v_{1i}^2} \times \sqrt{\sum_{i=1}^{n} v_{2i}^2}}$$

Interpretation of Values:

• 1.0: Vectors point in exactly the same direction (angle = 0°)
• 0.0: Vectors are orthogonal/perpendicular (angle = 90°)
• -1.0: Vectors point in exactly opposite directions (angle = 180°)
• Values between these indicate intermediate angular relationships

Key Properties:

1. Scale Invariance: Multiplying either vector by a positive scalar does not change the similarity value
2. Bounded Range: The result is always in the range [-1, 1]
3. Symmetry: The similarity between v₁ and v₂ equals the similarity between v₂ and v₁

Your Task: Write a Python function that computes the angular similarity between two vectors. The function should:

• Calculate the dot product of the two vectors
• Compute the magnitude (Euclidean norm) of each vector
• Return the normalized dot product, rounded to three decimal places

v1 = [1.0, 2.0, 3.0]
v2 = [2.0, 4.0, 6.0]

1.0

v1 = [1.0, 0.0]
v2 = [0.0, 1.0]

0.0

v1 = [1.0, 1.0, 1.0]
v2 = [-1.0, -1.0, -1.0]

-1.0