Parametric Lines & Planes

[sponsoredwalk]

Hi I'm reading Lang's Intro to Linear Algebra Page's 30 -36 and I'd just like some input on whether I understand this or not. Basically my problem is with Planes but the parametric eq. thing is just clarification.

Parametric Representation of a Line​

Basically, if you have two n-tuples (n=2 for convenience!)

(a,b) = A and (p,q) = P

We form the equations:

X(t) = P + tA <==> (x,y) = (p,q) + t(a,b)

with;

x = p + ta
y = q + tb (times -(a/b))
-----------
x = p + ta
-(a/b)y = -(a/b)q - ta
---------------------
x - (a/b)y = -(a/b)q

-(b/a)x + y = q

$$y \ = \ \frac{b}{a} \cdot x \ + \ q$$

This is forming the eq. of the line going from point P in the direction of A.

If you want to go from point A in the direction of P the equation becomes

X = A + tP

if you want to go in the opposite direction in either of the equations you just give t a negative value.

If you want to go just from point A to point P let:

0 ≤ t ≤ 1

and let P = P - A

so that X(t) = A + tP = A + t(P - A)

i.e. "t" takes on a value less than 1 so you multiply, say, 0.04 by all the values (P - A) represents in the (x,y) dimensions before going on.

Planes​

The idea is to find the equation of the plane in R³ that passes through some random point P by forming a located vector $$\overline{PX}$$, where X is the set of all points surrounding P, and then taking the dot product with some other located vector $$\overline{ON}$$ that is perpendicular to $$\overline{PX}$$.For some located vector $$\overline{ON}$$ if we want to find the plane that passes through P we'll dot product it:

(X - P) • (N - O) = 0

(X - P) • N = 0

X•N - P•N = 0

X•N = P•N

Okay, obviously O is the origin and can be ignored.

I'm thinking that N = (N - O) = $$\overline{ON}$$ can be done all the time,
no matter where the plane is to simplify the algebra.

It's like point N determines the angle/direction of the plane with respect to the origin & point P determines the height up or down. Is that correct? On page 34 of the book the picture of the plane could go up and down the N arrow.

Still, X•N = P•N doesn't seem intuitive to me, is there a way to get it?

Like, you're dot producting the variables X = (x,y,z) with a point not on the plane and that's supposed to be equal to the original point that the plane passes through dot producted with this point not on the plane you're constructing

 

[Noxide]

Just want to say congratz on understanding parametric representations of lines, and excellent choice on using a Springer text. Don't have time to check your plane work, but will hopefully get to it later tonight!

 

[tmccullough]

Both of those quantities are the minimum distance from the origin to the plane, meaning the distance from the origin to the plane traveling in the direction of the normal vector (actually times the length of the normal vector, which is typically chosen to be norm 1 for simplicity).

To see this geometrically, the dot product measures the component of the vector (pictures points in $\mathbb{R}^3$ as vectors pointing from the origin to the location of the point itself - there's always duality). This will be the same for every point on the plane - namely the minimum distance between the origin and the plane.

This same characterization works for the same reason for $n-1$-dimensional subspaces in $\mathbb{R}^n$, just with $n$ variables. It's actually equivalent to the typical characterization for lines in $\mathbb{R}^2.$