Let $T$ be the smallest positive real number such that the tangent to the hellix $$\cos t i + \sin t j + t/\sqrt{2} k$$ at $t = T$ is orthogonal to the tangent at $ t = 0$. Then, the line integral of $$F = x j - y i $$along section of the helix from $t = 0$ to $t = T$ is
Answer is $2$ but I can't understand how it came. Can any one help me please!!!
The tangent $(-\sin(T),\cos(T),1/\sqrt2)$ at $t = T$ being orthogonal to the tangent $(0,1,1/\sqrt 2)$ at $t = 0$ yields $\cos(T) = -1/2$, or $T = 2\pi/3$. Then unless I've made a mistake somewhere, the integral should be $$\int_0^{2\pi/3} \sin^2(t) + \cos^2(t) \, dt = 2\pi/3.$$