How to Find an Explicit Description of a Plane from an Implicit Equation?

[hogrampage]

I understand how to find an implicit description if given the span of, say, two vectors. How do I go about finding an explicit description of a plane as the span of two vectors? For example, where would I start if the plane equation was:

3x+2y-z = 0

Thanks!

 

[micromass]

What do you mean with "explicit description"? What is it of the plane that you would like to know?

 

[hogrampage]

i.e. Describe the plane as the span of a two-vector set.

 

[Erland]

i.e. Describe the plane as the span of a two-vector set.

You can only do that if the plane passes through the origin. Otherwise, it is such a span plus some constant vector.

But in your example, the plane passes through the origin, and the simplest way to find two vectors spanning the plane is to solve for one variable and put the others as parameters, say:
z=3x+2y, which leads to

$x=s$, $y=t$, $z=3s+2t$, or

$[x\,\, y\,\, z]^T=[s\,\, t\,\, 3s+2t]^T=s[1\, \,0\,\, 3]^T+t[0\,\,1\,\,2]^T$.

 

[hogrampage]

Okay, that is what I was thinking, but wasn't positive.

Thank you