Find the cosine of the angle between the curves $\langle 0,t^2,t \rangle$ and $\langle cos\left(\pi \frac{t}{2}\right)$,$sin \left(\pi \frac{t}{2} \right)$,$t \rangle$ where they intersect.
To solve this I first found the point of intersection by setting each of the respective components equal to each other.
$0 = cos \left(\frac{\pi u}{2}\right)$ solving this equation gives $1=u$
$t^2 = sin \left(\frac{\pi u}{2} \right)$ solving this equation gives $1=u$
$t = u$ solving this equations gives $1=u$
Putting this together I can see that my point of intersection is $\left( 0,1,1\right)$ Now to find the cosine of the angle between those curves I know I need the derivatives of those two vectors:
$\langle 0,2t,1 \rangle$ and $\langle -sin\left(\frac{\pi}{2}\right), cos\left(\frac{\pi}{2}\right), 1 \rangle$
From here, though, I'm not sure how to calculate the cosine of the angle between the two curves. Any hints?
Best Answer
The dot product of two vectors, u, v, is give by $u\cdot v= |u||v|cos(\theta)$ where $\theta$ is the angle between the two vectors. At t= 1, the two tangent vectors are <0, 1, 1> and <-1, 0, 1>.