Represent the line segment from P to Q by a vector-valued function and by a set of parametric equations P(−3, −6, −1), Q(−1, −9, −8).

Find the area of the parallelogram with vertices A(-3, 0), B(-1, 5), C(7, 4), and D(5, -1).

Proving Symmetry with Parametric Function
I'm trying to brush up on some math for my latest Computer Science class. We are given a shape (curve) in parametric form. How can we show that this is symmetric (or not) about the x-axis or y-axis?
If we have an equation in explicit form like: $y=mx+b$
Then to show that this line is symmetric about some axis, we can set either $x=-x$ or $y=-y$ and show if the equation remains the same. However, if we have an equation in parametric form, such as:
$x(t)=sin(t),y(t)=cos(t)$
Then how do we go about showing if this resulting shape is symmetric or not? Is there a way to do so without converting back to the explicit form?

For ${\displaystyle f\left(t\right)=(\frac{\ln t}{e^t},\frac{e^t}{t})}what\; is\; the\; distance\; between$ {\displaystyle f\left(1\right)}$$ and ${\displaystyle f\left(2\right)}$

What is the distance between the following polar coordinates?:
${\displaystyle (3,\frac{17\pi }{12}),(5,\frac{7\pi }{8})}$

For ${\displaystyle f\left(t\right)=(\ln t-e^t,\frac{t^2}{e^t})}$ what is the distance between f(2) and f(4)?

How do you write the parametric equations represent the ellipse given by ${\displaystyle \frac{x^2}{9}+\frac{y^2}{81}=1}$

How do you rewrite the two parametric equations’ as one Cartesian equation $x=\sec (t),y=\tan (t),-\frac{\pi }{2}<t<\frac{\pi }{2}?$