Vector Inner Product Computation

The inner product (commonly known as the dot product) is one of the most fundamental operations in linear algebra and serves as a cornerstone for countless algorithms in machine learning, physics, and computer graphics.

Given two vectors u = [u₁, u₂, ..., uₙ] and v = [v₁, v₂, ..., vₙ] of equal length n, the inner product is defined as the sum of the element-wise products of their corresponding components:

$$\langle u, v \rangle = \sum_{i=1}^{n} u_i \cdot v_i = u_1 v_1 + u_2 v_2 + \cdots + u_n v_n$$

Geometric Interpretation: The inner product has a beautiful geometric meaning. For two vectors, it equals the product of their magnitudes multiplied by the cosine of the angle between them:

$$\langle u, v \rangle = |u| \cdot |v| \cdot \cos(\theta)$$

This relationship makes the inner product invaluable for measuring:

• Similarity between vectors (higher values indicate more aligned directions)
• Orthogonality (inner product equals zero when vectors are perpendicular)
• Projections of one vector onto another

Your Task: Write a Python function that computes the inner product of two 1D NumPy-compatible arrays. The function should take two vectors as input and return their inner product as a single numeric value.

Note: You may assume both input vectors always have the same length (equal dimensions).

vec1 = [1, 2, 3], vec2 = [4, 5, 6]

32

vec1 = [2, 3], vec2 = [4, 5]

23

vec1 = [1, 2, 3, 4, 5], vec2 = [5, 4, 3, 2, 1]

35