which of the space curves above describes the vector - valued function: r(t)=⟨cos t,sin t,cos t sin 12t⟩…

which of the space curves above describes the vector - valued function: r(t)=⟨cos t,sin t,cos t sin 12t⟩? enter the number of the correct figure using one of the numerical labels 1 through 6.

which of the space curves above describes the vector - valued function: r(t)=⟨cos t,sin t,cos t sin 12t⟩? enter the number of the correct figure using one of the numerical labels 1 through 6.

Answer

Explanation:

Step1: Analyze $x = \cos t$ and $y=\sin t$

The parametric - equations $x = \cos t$ and $y=\sin t$ represent a circle in the $xy$ - plane since $x^{2}+y^{2}=\cos^{2}t+\sin^{2}t = 1$.

Step2: Analyze $z=\cos t\sin12t$

The $z$ - component $z = \cos t\sin12t$ oscillates as $t$ varies. The factor $\cos t$ has a period of $2\pi$ and $\sin12t$ has a period of $\frac{2\pi}{12}=\frac{\pi}{6}$. The product will cause the curve to move up and down along the $z$ - axis while the $x$ and $y$ components trace out a circle.

Answer:

(Without seeing the numbered figures, I can't give a specific number. But the curve should be a circular - like curve that oscillates in the $z$ - direction. If you can provide the figure numbers, I can give a more specific answer.)