Question

(a) Find a vector-parametric equation r⃗ 1(t)=⟨x(t),y(t),z(t)⟩r→1(t)=⟨x(t),y(t),z(t)⟩ for the shadow of the circular cylinder x2+z2=5x2+z2=5 in the xzxz-plane. Shadow: r⃗ 1(t)=r→1(t)= for 0≤t≤2π0≤t≤2π. (b) Find a vector-parametric equation for intersection of the circular cylinder x2+z2=5x2+z2=5 and the plane 3x+2y+8z=13x+2y+8z=1. Intersection: r⃗ 2(t)=r→2(t)= for 0≤t≤2π0≤t≤2π.

Answer 1

Answer: (a) r1(t) = <2cost , 0 , 2sint>

(b) <2cost , (1 - 12sint - 10cost)/8 , 2sint>

Step-by-step explanation:

x2+z2=4

a)

Now, in the xz plane, we know that y = 0...

So, x^2 + z^2 = 4 will simply be a circle centered at (0,0)..

This can be easily parameterized as

x = 2cos(t)

z = 2sin(t)

So, the required parameterization is :

r1(t) = <2cost , 0 , 2sint>

b)

Cylinder : x^2 + z^2 = 4

Plane : 5x+8y+6z=1

Easily enough, the x^2 + z^2 = 4 can again be parameterized as

x = 2cost , z = 2sint

With this, we can find y using plane equation...

5x+8y+6z=1

5(2cost) + 8y + 6(2sint) = 1

8y = 1 - 12sint - 10cost

y = (1 - 12sint - 10cost)/8

So, the parameterization is :

<2cost , (1 - 12sint - 10cost)/8 , 2sint>

Answer 2

(a). Parametric equation is, $r_{1}(t) = <2cost , 0 , 2sint>$

(b). Parametric equation is, $r_{2}(t)=<2cost , (1 - 12sint - 10cost)/8 , 2sint>$

Parametric equation:

It is a type of equation that employs an independent variable called a parameter .

                $x^{2} +y^{2}=r^{2}$

Parametric equation is,  $x=rcost,y=rsint$

(a). Now, in the XZ- plane, we know that y = 0

So, $x^2 + z^2 = 4$ will simply be a circle centered at (0,0)

This can be easily parameterized as

           $x = 2cos(t)$

           $z = 2sin(t)$

So, the required parameterization is :

           $r_{1}(t) = <2cost , 0 , 2sint>$

(b). Given that,  equation of Cylinder : $x^2 + z^2 = 4$

and equation of Plane : 5x+8y+6z=1

the $x^2 + z^2 = 4$ can again be parameterized as

   $x = 2cost , z = 2sint$

Substituting value of x and z in equation 5x+8y+6z=1

$5(2cost) + 8y + 6(2sint) = 1$

$8y = 1 - 12sint - 10cost$

$y = \frac{(1 - 12sint - 10cost)}{8}$

So, the parametric equation  is :

              $<2cost , (1 - 12sint - 10cost)/8 , 2sint>$

Learn more about the Parametric equation here:

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