Yes.Math10 said:Homework Statement
Find the curvature of r(t)=2ti+2tj+k.
Homework Equations
None.
The Attempt at a Solution
The answer is 0 in the book.
I know the formula for curvature is k(t)=abs(r'(t)xr"(t))/abs(r'(t))^3. I know that r'(t)=2i+2j and r"(t)=0, so r'(t)xr"(t)=0?
r'(t)r''(t) = 0, which you already found. What is |r'(t)|? Note that this means the magnitude or length of r'(t).Math10 said:How to find r'(t)xr"(t) and r'(t)^3?
Math10 said:So how do I find abs(r'(t))? That's where I got stuck.
The problem is that Math10 quoted the formula using abs() instead of modulus.SteamKing said:The notation |r'(t)| does not mean the absolute value of r'(t), it means to find the magnitude of the vector r'(t). Think Pythagoras.
Three of us have told you that |r'(t)| does not mean "absolute value". It means "magnitude" or "length" of the vector r'(t).Math10 said:So how do I find abs(r'(t))? That's where I got stuck.
The formula for finding the curvature of a parametric curve is given by K(t) = ||r'(t) x r''(t)|| / ||r'(t)||^3, where r(t) = xi(t) + yj(t) + zk(t) is the parametric equation of the curve.
The tangent vector of a parametric curve is given by r'(t), while the normal vector is given by r''(t) x r'(t). These can be found by taking the derivatives of the parametric equations for x(t), y(t), and z(t).
Yes, the curvature of a parametric curve can be negative. This indicates that the curve is bending in the opposite direction of the positive curvature. A negative curvature can also be referred to as "curvature of opposite sign."
The parameter t does not directly affect the curvature of a parametric curve. However, the values of t do determine the point on the curve for which the curvature is being calculated. Different values of t will result in different tangent and normal vectors, ultimately affecting the curvature at that point.
Yes, there are various ways to visualize the curvature of a parametric curve. One way is to plot the tangent and normal vectors at different points along the curve. Another way is to plot the osculating circle, which is a circle that best fits the curve at a particular point, with a radius equal to the inverse of the curvature at that point.