How to Mathematically describe a plane?

[HMPARTICLE]

The linear combinations of V = (1,1,0) and W = (0,1,1) fill a plane.
My question is how do i describe that plane? (not geometrically).

 

[WWGD]

This plane is the set { s(1,1,0)+t(0,1,1): s,t Reals}={(s, s+t, t)} . Is that what you are looking for?

 

[HMPARTICLE]

Yes! I understand what the question is asking now! I'm making that transition between A-level and UG.

Thanks WWGD.

 

[HallsofIvy]

Notice that the cross product is <1, 1, 0> and <0, 1, 1> is <1, -1, 1> and, since this plane contains the origin, it could also be described by the single equation x- y+ z= 0.

 

[blue_raver22]

To mathematically describe a plane, we can use the equation of a plane in vector form:

r = r0 + sv + tw

where r is a position vector on the plane, r0 is a fixed point on the plane, v and w are two non-parallel vectors that span the plane, and s and t are any real numbers.

In this case, the vectors V = (1,1,0) and W = (0,1,1) are non-parallel and span the plane. So, we can use them to describe the plane as:

r = r0 + s(1,1,0) + t(0,1,1)

To find the fixed point r0, we can choose any point on the plane and substitute its coordinates into the equation. For example, if we choose the point (1,1,0), we get:

(1,1,0) = r0 + s(1,1,0) + t(0,1,1)

Since s and t can be any real numbers, we can choose them to be 0, which gives us:

r0 = (1,1,0)

So, the equation of the plane can be written as:

r = (1,1,0) + s(1,1,0) + t(0,1,1)

This equation can also be written in component form as:

x = 1 + s + t
y = 1 + s
z = t

This is the mathematical description of the plane formed by the linear combinations of V and W. It represents all the points on the plane in terms of two parameters, s and t, which can take on any real values.