1. What is Inner Product?

The inner product (also called dot product) of two vectors is a scalar value that measures their similarity and angle between them. In 2D space, it's calculated as the sum of the products of corresponding components.

2. How Does the Calculator Work?

The calculator uses the inner product formula:

Where:

• \( u_x \) — x-component of vector u
• \( u_y \) — y-component of vector u
• \( v_x \) — x-component of vector v
• \( v_y \) — y-component of vector v

Explanation: The inner product combines both the magnitudes of the vectors and the cosine of the angle between them.

3. Importance of Inner Product

Details: Inner products are fundamental in vector analysis, physics, and machine learning. They're used to determine angles between vectors, test for orthogonality, and in projections.

4. Using the Calculator

Tips: Enter all four vector components (u_x, u_y, v_x, v_y) as real numbers. The calculator will compute their inner product.

5. Frequently Asked Questions (FAQ)

Q1: What does the inner product represent geometrically?
A: It equals the product of the vectors' magnitudes and the cosine of the angle between them: \(\|\mathbf{u}\| \|\mathbf{v}\| \cos \theta\).

Q2: What does a zero inner product mean?
A: Vectors are orthogonal (perpendicular) to each other when their inner product is zero.

Q3: Can inner product be negative?
A: Yes, when the angle between vectors is greater than 90 degrees.

Q4: How is this different from cross product?
A: Inner product gives a scalar, while cross product gives a vector (in 3D). Inner product measures similarity, cross product measures perpendicularity.

Q5: What are applications of inner product?
A: Used in physics (work calculation), computer graphics (lighting), machine learning (kernel methods), and more.