# Find Curvature of r(t)=2ti+2tj+k

• Math10
So you need to find the length of the vector 2i + 2j. Can you use Pythagoras's theorem to find the length of this vector?In summary, we are trying to find the curvature of the vector function r(t)=2ti+2tj+k using the formula k(t)=abs(r'(t)xr"(t))/abs(r'(t))^3. After determining that r'(t)=2i+2j and r"(t)=0, we need to find the magnitude or length of r'(t) by using Pythagoras's theorem. This will allow us to solve for the curvature.
Math10

## Homework Statement

Find the curvature of r(t)=2ti+2tj+k.

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? How to find r'(t)xr"(t) and r'(t)^3?

Math10 said:

## Homework Statement

Find the curvature of r(t)=2ti+2tj+k.

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?
Yes.
Math10 said:
How to find r'(t)xr"(t) and r'(t)^3?
r'(t)r''(t) = 0, which you already found. What is |r'(t)|? Note that this means the magnitude or length of r'(t).

So how do I find abs(r'(t))? That's where I got stuck.

Math10 said:
So how do I find abs(r'(t))? That's where I got stuck.

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.

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.
The problem is that Math10 quoted the formula using abs() instead of modulus.
It should be ##k(t)=\frac{|\dot{\vec r}(t)\times \ddot {\vec r}(t)|}{|\dot{\vec r}(t)|^3}##

Math10 said:
So how do I find abs(r'(t))? That's where I got stuck.
Three of us have told you that |r'(t)| does not mean "absolute value". It means "magnitude" or "length" of the vector r'(t).

## 1. What is the formula for finding the curvature of a parametric curve?

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.

## 2. How do you find the tangent and normal vectors of a parametric 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).

## 3. Can the curvature of a parametric curve be negative?

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."

## 4. How does the parameter t affect the curvature of a parametric curve?

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.

## 5. Is there a way to visualize the curvature of a parametric curve?

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.

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