10+2 Vector Algebra: Dot, Cross Products & Gradient Fields
Vectors are mathematical entities possessing both magnitude and direction. In engineering physics and 3D computer graphics, vectors form the foundation of force analysis, ray tracing, and spatial transformations.
1. The Dot Product (Scalar Product)
The scalar product between two vectors $\vec{a}$ and $\vec{b}$ measures projection and alignment:
Dot Product Formula
Orthogonality Condition
Two non-zero vectors are perpendicular if and only if their dot product equals zero ($\vec{a} \cdot \vec{b} = 0$). This property is used constantly in game engine lighting shaders to calculate Lambertian diffuse reflection.
2. The Cross Product (Vector Product)
The vector product creates a vector perpendicular to both input vectors following the right-hand rule:
Cross Product Determinant
3. Interactive Vector Field Simulation
Explore dynamic vector fields, vortex curls, and field divergence in the TechNHIT Visual Lab:
Vector Field Curl & Divergence in 2D Space
SVG // HIGH-AVAILABILITY TOPOLOGYHAProxy VIP10.10.10.100
── 10GbE ──▶
Ollama-01RTX 3090 #1
Ollama-02RTX 3090 #2
── NVMe-oF ──▶
ZFS PoolRAIDZ2 8TB
4. Vector Calculations in NumPy
NumPy Vector Verification
Python 3
$
python3 -c 'import numpy as np; a = np.array([1, 2, 3]); b = np.array([4, 5, 6]); print(\