Parametric Description of a Plane

[mill]

I read the definition that a plane is a point and two vectors with the equation being plane sum = {OP + tv + sw} where v and w are vectors and t and s are real numbers. This is called the parametric description of the plane. I haven't seen the equation in this form before though.

Can someone explain what these values stand for/how to use this equation or direct me to a page that explains it? I just see the regular plane equation when I google this.

 

[Simon Bridge]

You are aware that a plane can be defined by three points that are not co-linear?
This form is just the same - using two pairs of points to form the two vectors.

Any two vectors $\vec v$ and $\vec w$ must lie in a common plane.
In fact, they can define a set of parallel planes.

The parametric equation is just the instructions to get to another point in the plane starting from where you are at.

i.e. If point $P$ is in the plane defined by the above vectors, then you can get from there to point $Q$, also in the pane, by starting out at $P$ and walking $t$ steps of size $v$ in the direction of $\vec v$, then turning to the direction of $\vec w$ and walking $s$ steps of size $w$ in that direction.

In maths that is: $Q=P+t\vec v + s\vec w$

Concrete example:$$\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}1\\-1\\2\end{pmatrix}+t\begin{pmatrix}3\\4\\0\end{pmatrix}+s\begin{pmatrix}-1\\0\\0\end{pmatrix}$$... tells you how to get to point (x,y,z) from point (1,-1,1) using (3,4,0) and (-1,0,0) as cardinal directions.

i.e. you want to get to (x,y)=(8,8), then t=2 and s=-1
so you must travel 10 units along the hypotenuse of the 3-4-5 triangle, then 1 unit parallel to the x-axis.

Notice that the plane in the example is parallel to the cartesian x-y plane, and have expressed the vectors in cartesian coordinates. I don't have to.

The directions done this way basically translate the (t,s) coordinates for positions on the plane to (x,y,z) cartesian coordinates ...

 

[mill]

You are aware that a plane can be defined by three points that are not co-linear?
This form is just the same - using two pairs of points to form the two vectors.

Any two vectors $\vec v$ and $\vec w$ must lie in a common plane.
In fact, they can define a set of parallel planes.

The parametric equation is just the instructions to get to another point in the plane starting from where you are at.

i.e. If point $P$ is in the plane defined by the above vectors, then you can get from there to point $Q$, also in the pane, by starting out at $P$ and walking $t$ steps of size $v$ in the direction of $\vec v$, then turning to the direction of $\vec w$ and walking $s$ steps of size $w$ in that direction.

In maths that is: $Q=P+t\vec v + s\vec w$

Concrete example - (x,y)=(1,-1)+t(3,4)+s(1,0) tells you how to get to point (x,y) from point (1,-1) using (3,4) and (1,0) as cardinal directions.

i.e. you want to get to (x,y)=(8,8), then t=2 and s=1
so you must travel 10 units along the hypotenuse of the 3-4-5 triangle, then 1 unit parallel to the x-axis.

Thanks. That cleared it up. I had never thought of Q like that, but it really helped.

 

[Simon Bridge]

No worries.
Similarly the parametric equation for a line is $Q=P+s\vec v$ and for a 3D volume you need three parameters.
It gets more fun when you use surfaces instead of planes - those are allowed to curve.