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Find parametric equations for the line through the point (0, 2, 2) that is parallel to the plane x + y + z = 4 and perpendicular to the line x = 1 + t, y = 2 − t, z = 2t.

Respuesta :

lhannearaujo

Answer:

The parametric equations follow bellow:

x=3t

y=2-t

z=2-2t

Step-by-step explanation:

From exercise you have the following data:

  • Point [tex](0,2,2)[/tex]
  • Plane  [tex]x+y+z=4[/tex] - parallel to the plane
  • Line [tex]x=1+t, y=2-t, z=2t[/tex] - perpendicular to the line

To find parametric equations consider the next steps.  

  • Step 1. Find vectors for the plane and the line:

Plane

[tex]x+y+z=4\\\\\\Then \\ vector = < 1,1,1>[/tex]

Line

[tex]x=1+t \\y=2-t\\ z=2t \\\\\\Then \\ vector = < 1,-1,2>[/tex]

  • Step 2. Find direction vector:

You should calculate the cross-product of the vectors of the plane and the line that you found in step 1. Therefore:

[tex]\left[\begin{array}{ccc}i&j&k\\1&1&1\\1&-1&2\end{array}\right][/tex]

[tex]2i+j-k-2j+i-k\\3i-j-2k[/tex]

Then, direction v=<3,-1,-2>

  • Step 3. Find the vector equation of the line

[tex]s(t)= point + t (crossproductresult}) \\\\s(t)= <0,2,2> + t <3,-1,-2>\\[/tex]

  • Step 4. Find parametric equations

[tex]x=0+3t=3t\\y=2-t\\z=2-2t[/tex]

Therefore, the parametric equations are:

x=3t

y=2-t

z=2-2t

Learn more about parametric equations here:

https://brainly.com/question/13072659

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