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Interactive 3D/Vectors in 3D
|u| = 1.526
|v| = 1.315
u·v = 0.760
θ = 67.8°
|u×v| = 1.858
→ u·→ v·→ u+v·→ u×v
Vector u (indigo)
uₓ1.2
u_y0.8
u_z0.5
Vector v (orange)
vₓ0.3
v_y1.0
v_z-0.8
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Dot product u·v = |u||v|cos(θ). Zero means orthogonal (perpendicular). Negative means obtuse angle (>90°).
Cross product u×v gives a vector perpendicular to both. Its magnitude = area of the parallelogram they form.

Vectors in 3D - Interactive Visualization

Vectors are the fundamental language of machine learning. Every data point, model weight, and gradient is a vector. This interactive visualization lets you manipulate two 3D vectors u and v with xyz sliders, instantly seeing their sum, cross product direction, angle between them, and the plane they span. When the dot product equals zero, the vectors are orthogonal - a concept at the heart of PCA, attention mechanisms, and regularization.

  • Adjust xyz components and see vector operations update instantly
  • Visualize u+v (sum), u×v (cross product), and the angle between them
  • Understand dot product u·v = |u||v|cos(θ) geometrically
  • See when two vectors are orthogonal (dot product = 0)
  • Foundation for understanding PCA, attention, and embeddings

Part of the EngineersOfAI Interactive 3D - free interactive visualizations covering every major concept in machine learning and AI engineering. Hover any element for a plain-English explanation. No code required.