How to plot circle, with it centered at a point in 3D in a plane of xyz equation? [duplicate]

Question

I can figure out how to plot this for calc 3.

Question:

a) Plot the circle of radius 3 centered at the point $\{-1, 1, 1\}$ in the plane whose xyz-equation is $2(x + 1) + 3(y - 1) + (z - 1) = 0$. Include in your plot a big enough piece of the plane to accommodate the circle.

My answer:

We are given:

• Radius = $3$
• Center = $\{-1, 1, 1\}$

Then:

$2(x + 1) + 3(y - 1) + (z - 1) = 0$ so

$2x + 2 + 3y - 3 + z - 1 = 0$ so

$2x + 3y + z = -2 + 3 + 1$ so

$2x + 3y + z = 2$

Thus: Normal = $\{2, 3, 1\}$

Equation of line:

$r(t) = \{-1, 1, 1\} + t \{2, 3, 1\}$

Equation of plane:

$2x + 3y + z = 2$ so

$\left(x + \frac{3}{2}y + \frac{z}{2}\right) = 1$

So we can find that the:

• x intercept = $1$
• y intercept = $\frac{2}{3}$
• z intercept = $2$

How can I plot this?

b) Here's a plot of a spiral in the xy-plane:

Clear[spiral, t];
spiral[t_] = {t Cos[2 t], t Sin[2 t]};
ParametricPlot[spiral[t], {t, 0, 3 Pi}, AxesLabel -> {"x", "y"}]

Use your answer to part a) above to help plot a true scale duplicate copy of this spiral on the plane with xyz-equation $2(x + 1) + 3(y - 1) + (z - 1) = 0$. Center your spiral at $\{-1, 1, 1\}$ and include in your plot a big enough hunk of the plane to accommodate the spiral.

How do I plot this?

The image is what I have tried.

Answer 1

I'll teach you how to do part 1; the method for part 2 is completely analogous.

Note that the equation for your plane is already in Hessian form; you now know that the plane passes through $(-1,1,1)$ and has the normal with components $\langle2,3,1\rangle$. If you start with the parametric equation of a circle of radius $3$ in the $x$-$y$ plane:

circ[t_] := {Cos[t], Sin[t], 0};

then you can derive the equation for the embedded circle as

newCirc[t_] = {-1, 1, 1} + RotationTransform[{{0, 0, 1}, {2, 3, 1}}][circ[t]];

To show that the circle derived fits the bill:

Show[ParametricPlot3D[newCirc[t], {t, 0, 2 π}], 
     Graphics3D[{Opacity[2/3], Hyperplane[{2, 3, 1}, {-1, 1, 1}], Sphere[{-1, 1, 1}, 0.03],
                 Arrow[Tube[{{-1, 1, 1}, {-1, 1, 1} + Normalize[{2, 3, 1}]}, 0.005]]}],
     PlotRange -> All]

As I said, part 2 is up to you; you just need to do a similar procedure for your spiral. You should be able to get a picture like this: