Putting equations in parametric form

Question

Im told that an object has the following trajectory:

$$ \begin{gather} T= \left\{ \begin{array}{ll} x^2 + y^2 &= 4\\ x + z &=3 \end{array} \right. \end{gather} $$

We are also told that the object makes a full tour around T.

I'm asked to put T in parametric form, but I don't understand how, since there is no $t$ in the equation.

And also, I don't understand what it means when they say that the object makes a full tour around T. Does that mean T is a circle?

Answer 1

In the $(xy) $ plane, the trajectory is the circle of center $(0,0) $ and radius $2$.

in the space, the object moves on a cylindrical surface defined by the parametric equations:

$$x=2\cos (t) $$ $$y=2\sin (t) $$ $$z=3-x=3-2\cos (t) $$

Answer 2

Attempting an interpretation:

Let me write $\theta$ instead of t:

$x = 2 cos(\theta)$ ; $y = 2 sin(\theta)$ : $z = 3 - 2 cos(\theta)$ .

In vector form:

$\vec{r}$ = $(0,0,3)$ + $2( cos(\theta), sin(\theta), - cos(\theta) )$.

$\vec{r} = (0,0,3) + \vec{r'}$

Let's look at the direction vector $\vec{r'}$:

$\vec{r'} = 2( cos(\theta), sin(\theta), - cos(\theta) )$.

Projection of path onto the $x -, y -$ plane is a circle, I.e. the object moves on a cylinder, radius = $2$. Cylinder axis parallel to $z$ - axis.

The $z$ component describes a $cos$ curve.

The projection of the path onto the $y-, z-$ plane is again a circle, centre at $y = 0$, $z = 3$, I.e. the object moves on a cylinder, radius = $2$. Cylinder axis parallel to $x$ - axis

The x component describes a $cos$ curve.

Comments welcome.