Tangent vector and unit tangent vector

 Goal:

1. Define tangent vector and unit tangent vector

Tangent vector and unit tangent vector

Let's remind that the derivative of a function at a point is the slope of the tangent line to the graph of this function at this point. In the case of a vector-valued function the derivative provides a tangent vector to the graph of this function.

Let's consider a vector-valued function and see what a tangent vector looks like on its graph. We have the vector-valued function r(t) = sinti + costj . Its derivative is r'(t) = costi-sintj. Let's substitute t = π/6 in both functions: 

Here is the graph of the function with the vectors r(π/6) r'(π/6):

Notice that the the tangent vector r'(π/6) is tangent to the circle at the corresponding point t =  π/6. This is an example of the tangent vector to the plane curve defined by the function r(t) = sinti + costj

Definition

Let C be a curve defined by a vector-valued function r, and assume that r'(t) exists when t = t₀. A tangent vector v at t = t₀.is any vector such that, when the tail of the vector is placed at point r(t₀) on the graph, vector v is tangent to curve C. Vector r'(t₀) is an example of a tangent vector at point t =  t₀.. Furthermore, assume that r'(t) ≠ 0. The principal unit tangent vector at t is defined to be:

 

The unit tangent vector is exactly what it sounds like: a unit vector that is tangent to the curve. To calculate a unit tangent vector, find the derivative r'(t). Second, calculate the magnitude of the derivative. The third step is to divide the derivative by its magnitude.

Example

Solution

Practice