Converting from Cartesian to Parametric form

[JFonseka]

[SOLVED] Converting from Cartesian to Parametric form

Homework Statement

Find a parametric vector equation of for the plane in R^3 having cartesian equation

4y + 5z = - 6

Homework Equations

None

The Attempt at a Solution

What I did was, first I turned the equation into 4x + 5y = -6, cause I'm more comfortable just treating equations like that.

Then I solved for y and got, y = (-4x -6)/5 = t

Hence x = (4t + 6)/5

and now y = t.

Then in parametric form it becomes:

(x, y) = (6/5, 0) + (4/5, 1)t

Is this right or wrong?

 

[genneth]

As it's a plane, you need two parameters to describe it. This is a standard transform -- the equations for a plane in cartesian form and parametric form are well known, and have easy to understand geometric ties. Understand how they both relate to the geometry, and it will be easy to swap between them.

 

[HallsofIvy]

Homework Statement

Find a parametric vector equation of for the plane in R^3 having cartesian equation

4y + 5z = - 6

Homework Equations

None

The Attempt at a Solution

What I did was, first I turned the equation into 4x + 5y = -6, cause I'm more comfortable just treating equations like that.

Unfortunately, it's not true now! Your original equation had y and z, not x and y! You can't just arbitrarily change it!

Then I solved for y and got, y = (-4x -6)/5 = t

Hence x = (4t + 6)/5

?? If y= (-4x-6)/5, then x is NOT (4y+6)/5. From 4x+ 5y= -6 (which is incorrect itself), solving for x gives x= (-5y-6)/4.

and now y = t.

Then in parametric form it becomes:

(x, y) = (6/5, 0) + (4/5, 1)t

Is this right or wrong?

What happened to z?? This is a plane in three dimensions. Since a plane is a two-dimensional object, parametric equations must involve two independent parameters.
You can do this: solve 4y+ 5z= -6 for either y or z: say, z= (-6-4y)/5. Now you can let y be the parameter: y= t, z= (-6-4t)/5.

Since your equation tells you nothing about x, x can be any number for all y and t: let x be the other parameter. Now what do you have?

 

[JFonseka]

I still don't get it, so Iwe have z = (-6 -4t)/5 now and I let x be equal to anything?

 

[Dick]

I still don't get it, so Iwe have z = (-6 -4t)/5 now and I let x be equal to anything?

As genneth and Halls said, you should expect to have two parameters. y= t, z= (-6-4t)/5, x=s is one way to write it.

 

[JFonseka]

Yea I managed to solve it a few hours ago, thanks for the replies guys!