Question

Prove that the curve a(t) = (cost, sin 2t, cos 2t) is regular on R and that it self-intersects at (1,0,1). Check the self-intersection part by using algebra and also by using Geofte

Answer 1

Answer:

The function a (t) is a vector function composed of the component functions $a_ {1} (t) = cost, a_ {2} (t) = sin2t$ and $a_ {3} (t) = cos2t$. How $a_ {1} (t), a_ {2} (t), a_ {3} (t)$ are infinitely derivable functions in R, so they are regular functions in R.

Now, for$t = 0$, you have to $a (0) = (cos (0), sin2 (0), cos2 (0)) = (1, 0, 1)$. How the functions $a_ {1} (t), a_ {2} (t), a_ {3} (t)$ are periodic functions with period $2 \pi,$ the vector function $a (t)$ will take the same point $(1, 0 , 1)$ at $t = 2n\pi, n = 0, 1, 2, 3, ...$ then the vector function is auto-intercepted

Step-by-step explanation: