Vector Guides
Clear, worked walkthroughs of the named unit vectors, the ones a course names in passing and then expects you to build. Every guide ends by normalizing something and checking the length is exactly 1.
Showing 6 of 6A unit normal vector has length 1 and points perpendicular to a surface or curve. Learn how to find one for both senses, with worked examples.
The unit tangent vector T̂ = r′(t)/‖r′(t)‖ points along the direction of travel on a curve. Learn how to find one, with two worked examples.
The TNB frame is three perpendicular unit vectors on a curve: T̂ along the path, N̂ into the bend, B̂ = T̂ × N̂. Build all three on a helix and check them.
Two ways to find the normal vector to a plane: read it off ax + by + cz = d, or cross two edge vectors from three points. Both worked, sign explained.
Polar unit vectors r̂ = ⟨cos θ, sin θ⟩ and θ̂ = ⟨−sin θ, cos θ⟩ change with position. See why, with a verify run and a Cartesian-to-polar example.
Component tables for cylindrical and spherical unit vectors, a verify run on r̂, θ̂, φ̂, and the θ/φ convention trap that breaks copied formulas.