Quadratic Equation Solver (ax² + bx + c = 0)
About Dot Product Calculator
The Vector Dot Product (Scalar Product $\vec{a} \cdot \vec{b}$) Calculator computes scalar projections, directional alignment, orthogonality tests ($\vec{a} \cdot \vec{b} = 0$), and angles between vectors ($\theta = \arccos(\frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|})$) for N-dimensional vectors.
How to Use Dot Product Calculator
Step 1
Enter components for Vector A and Vector B separated by commas or spaces.
Step 2
Review computed dot product scalar and angle between vectors.
Step 3
Inspect cosine similarity metric and projection length.
Step 4
Click "Copy Dot Product".
Practical Use Cases for Dot Product Calculator
3D Computer Graphics Lambertian Diffuse Lighting
Calculate light intensity hitting 3D polygons by taking the dot product between the surface normal $\vec{N}$ and light direction vector $\vec{L}$.
Machine Learning Cosine Similarity & Vector Search
Measure semantic text similarity between high-dimensional embedding vectors using normalized dot products.
Input & Output Examples
Calculating Dot Product & Angle for Two 3D Vectors
Vector a: `[2, 3, -1]`, Vector b: `[4, -2, 5]`
Dot Product: `(2×4) + (3×(-2)) + (-1×5) = 8 - 6 - 5 = -3` | Angle: `96.88° (1.691 rad)` | Orthogonal: `No`
Key Features & Performance
- ✓Computes: Algebraic Dot Product ($\sum a_i b_i$), Geometric Dot Product ($|a||b|\cos\theta$), Angle in Degrees/Radians, Scalar Projection.
- ✓Cosine Similarity score generator ($-1.0$ to $+1.0$).
- ✓Supports arbitrary dimensions from 2D up to 1,024-dimension ML embedding vectors.
- ✓100% Client-Side memory execution.
- ✓1-Click Copy solution proof.
Key Terminology & Definitions
Dot Product ($\vec{a} \cdot \vec{b}$)
An algebraic operation that takes two equal-length coordinate vectors and returns a single scalar number representing their directional alignment.
Orthogonality
Two non-zero vectors are perpendicular (orthogonal) if and only if their dot product equals zero ($\vec{a} \cdot \vec{b} = 0$).