Tangent vectors, or how to go straight when you are on a bender. Copyright 2008 by Evans M. Harrell II. MATH 2401 - HarrellIn our previous episode: 1. Vector functions are curves. The algebraic side of the mathematian’s brain thinks about vector functions. The geometric side sees curves.In our previous episode: 1. Vector functions are curves. 2. Don’t worry about the basic rules of calculus for vector functions. They are pretty much like the ones you know and love.Tangent vectors – the derivative of a vector functionTangent vectors  Think velocity!  Tangent lines  Tell us more about these!Tangent vectors  The velocity vector v(t) = r′(t) is tangent to the curve – points along it and not across it.Example: spiralCurves  Position, velocity, tangent lines and all that: See Rogness applets (UMN) at http://www.math.umn.edu/~rogness/multivar/calc2demos/curves.htmlTangent vectors  Think velocity!  Tangent lines  Approximation and Taylor’s formula  Numerical integration  Maybe most importantly – T is our tool for taking curves apart and understanding their geometry.Example: helixExample: HelixUnit tangent vectors  Move on curve with speed 1.  T(t) = r′(t)/ |r′(t)|  Only 2 possibilities, ± T.Example: ellipseExample: ellipseExample: ellipseExample: spiralExample: spiralExample: HelixCurves  Even if you are twisted, you have a normal!Normal vectors  A normal vector points in the direction the curve is bending.  It is always perpendicular to T.  What’s the formula?……………Normal vectors  N = T′/||T′||.  Unless the curve is straight at position P, by this definition N is a unit vector perpendicular to T. Why?Example: spiral in 3DExample: spiral in 3DExample: spiral in 3D Interpretation. Notice that the normal vector has no vertical component. This is because the spiral lies completely in the x-y plane, so an object moving on it is not accelerated vertically.Example: HelixExample: HelixExample: Helix Interpretation. Notice that the normal vector again has no vertical component. If a particle rises in a standard helical path, it does not accelerate upwards of downwards. The acceleration points inwards in the x-y plane. It points towards the central axis of the helix.Example: solenoidExample: solenoid Scary, but it might be fun to work it out!Arc length  If an ant crawls at 1 cm/sec along a curve, the time it takes from a to b is the arc length from a to