∇
VECTOR Calc 1–4
4d Streak
120 XP
Mode A: Guided Applet
x₀ = 1.00| f(x₀) = 0.750| f'(x₀) = 0.250
Dynamic Parameters
Target Point x₀ (x0x_0)1.0
Epsilon Corridor ε (ε\varepsilon)0.50
Delta Window δ (δ\delta)0.35
Visual Overlays & Layers
Chapter 1 Theory

3D Vectors, Dot & Cross Products & Geometry of Planes

Orthogonality, torque vectors, and normal vectors determining 3D planes

Interact with 3D vectors u and v to observe their scalar dot product u·v and the perpendicular cross product u×v.

1. The Geometric Meaning of Dot & Cross Products

The dot product u·v is scalar projection measuring alignment (zero when perpendicular). The cross product u×v produces a vector perpendicular to both, whose magnitude equals the parallelogram area.

n=u×v,n⋅(r−r0)=0\mathbf{n} = \mathbf{u} \times \mathbf{v}, \qquad \mathbf{n} \cdot (\mathbf{r} - \mathbf{r}_0) = 0

Quick Reference & Cheat Sheet

Plane Equation from Normal Vector
a(x−x0)+b(y−y0)+c(z−z0)=0a(x - x_0) + b(y - y_0) + c(z - z_0) = 0
Every displacement vector on the plane is orthogonal to normal vector n = <a,b,c>.