Angle between vector and tangent vector

[Prof. 27]

Homework Statement

My problem is:

For the logarithmic spiral R(t) = (e^t cost, e^t sint), show that the angle between R(t) and the tangent vector at R(t) is independent of t.

Homework Equations

N/A

The Attempt at a Solution

The tangent vector is just the vector that you get when you take the derivative of each element of the vector so:

R(t) = (e^t cost, e^t sint)
R'(t) = (e^t*cost-e^t*sint, e^t*sint + e^t*cost)

First I tried to show that the dot product was zero using the multiply the x's and y's then add method. This would imply that the angle between the vector and tangent vector was always ninety degrees; unfortunately as can be seen from an inspection of the two vectors, the dot product is e^2t.

Then I went the long route and used the dot product formula (a dot b = ||a||*||b||*cos(@)) to try to calculate a single angle @ that held constant between them. This was also a bust.

e^2t = ||R(t)|| * ||R'(t)||* cos(@) = sqrt(e^2t) * sqrt(2*e^2t) * cos(@)
= sqrt(2*e^4t) * cos(@).

It is easy to then see that @ = cos^-1(e^2t / sqrt(2*e^(4t)), which is obviously still dependant on t.

Any ideas?

 

[ehild]

Homework Statement

My problem is:

For the logarithmic spiral R(t) = (e^t cost, e^t sint), show that the angle between R(t) and the tangent vector at R(t) is independent of t.

Homework Equations

N/A

The Attempt at a Solution

The tangent vector is just the vector that you get when you take the derivative of each element of the vector so:

R(t) = (e^t cost, e^t sint)
R'(t) = (e^t*cost-e^t*sint, e^t*sint + e^t*cost))##

First I tried to show that the dot product was zero using the multiply the x's and y's then add method. This would imply that the angle between the vector and tangent vector was always ninety degrees; unfortunately as can be seen from an inspection of the two vectors, the dot product is e^2t.

Then I went the long route and used the dot product formula (a dot b = ||a||*||b||*cos(@)) to try to calculate a single angle @ that held constant between them. This was also a bust.

e^2t = ||R(t)|| * ||R'(t)||* cos(@) = sqrt(e^2t) * sqrt(2*e^2t) * cos(@)
= sqrt(2*e^4t) * cos(@).

It is easy to then see that @ = cos^-1(e^2t / sqrt(2*e^(4t)), which is obviously still dependant on t.

Any ideas?

What is $\sqrt{2e^{4t}}$?

 

[LCKurtz]

Homework Statement

My problem is:

For the logarithmic spiral R(t) = (e^t cost, e^t sint), show that the angle between R(t) and the tangent vector at R(t) is independent of t.

Homework Equations

N/A

The Attempt at a Solution

The tangent vector is just the vector that you get when you take the derivative of each element of the vector so:

R(t) = (e^t cost, e^t sint)
R'(t) = (e^t*cost-e^t*sint, e^t*sint + e^t*cost)

First I tried to show that the dot product was zero using the multiply the x's and y's then add method. This would imply that the angle between the vector and tangent vector was always ninety degrees; unfortunately as can be seen from an inspection of the two vectors, the dot product is e^2t.

Then I went the long route and used the dot product formula (a dot b = ||a||*||b||*cos(@)) to try to calculate a single angle @ that held constant between them. This was also a bust.

e^2t = ||R(t)|| * ||R'(t)||* cos(@) = sqrt(e^2t) * sqrt(2*e^2t) * cos(@)

Note that $\sqrt{e^{2t}} = e^t$. Then recheck your $\|\vec R'\| = \sqrt{2e^{2t}}$. Get it right and that last line will do it for you.