Vector Functions (Part 2) Flashcards
What is the unit tangent vector of a vector function r(t)?
It is the quotient between the tangent vector of r(t) and its norm
What is the unit normal vector?
It is the quotient between the derivative of the unit tangent vector and its norm.
What is the binormal vector?
It is the cross product between the unit tangent vector and the unit normal vector.
What can we assure about a vector function which norm is constant for all values of t?
That its derivative is normal to the vector function.
What do the unit tangent, normal and binormal vectors tell you?
They help describe the notion of a 3-D curve.
The tangent vector tells you the direction in which you’re going to
The normal one tells you how the tangent vector changes, that is, how the direction in which the particle is going to is changing
The binormal
What’s the normal plane at a point P = r(to)?
It is the plane that is parallel to both the normal and binormal unit vectors.
In other words, it passes through the point P and it’s orthogonal to the unit tangent vector.
What’s the rectifying plane at a point P = r(to)?
It is the plane that is parallel to both the tangent and binormal unit vectors.
In other words, it passes through the point P and it’s orthogonal to the unit normal vector.
What is the curvature of a curve and how to get it?
It can be gotten by taking the norm of the second derivative of a vector function parametrized by arc-length.
This measures the flexion of the curve.
How to get the curvature of a vector function that is not parametrized by arc-length?
by taking the norm of the derivative of the unit tangent vector and dividing it by the norm of the derivative of the vector function.
By taking the norm of the cross product between r’(t) and r’‘(t) and dividing it by the cube of the norm of r’(t).
What is the curvature of a function y = f(x) that is twice differentiable?
|f’‘(x)| / [1 + (f’(x))^2]^3/2
What is the curvature of circumference at a point P of a plane curve with curvature different than zero?
It is the circumference that is
1) tangent to the curve at P
2) it has the same curvature of the curve at P
3) it is located in the region of the plane that is concave to the curve
What the radius of curvature equal to?
Ro (t) = 1/k(t)
