can someone help please:
Find what kind of curves are given by the following representations and draw the curves:
1. r(t) = (2t-5, -3t+1, 4)
2. r(t) = (0, -cos(t), 3sin(t))
Sure. In the first case
$\displaystyle x = 2t -5,\;\;y= -3t+1,\;\;z=4$
z isn't changing, x and y are. Eliminating t gives
$\displaystyle \frac{x+5}{2} = \frac{y-1}{-3}$ a straight line.
In the second, $\displaystyle x = 0$, it's not changing and
$\displaystyle y = - \cos t,\;\; z = 3 \sin t$
Eliminating t using the fact that $\displaystyle \cos^2 t + \sin^2 t= 1$ gives
$\displaystyle y^2 + \frac{z^2}{3^2} = 1$, the formula for an ellipse in the yz plane. Hope that helps.