How Do Lines and Planes Differ in 3D Geometry?

[lonewolf219]

Not quite understanding what creates a plane, and what creates a line in 3D. Are vectors points in the plane with a line drawn from the origin to the point? Are position vectors vectors drawn between two vectors? If a plane is a vector and a point, how is that different from the description of a line? Not grasping the concepts here...

 

[Number Nine]

Vectors are none of those things. Vectors are just objects that obey certain rules (see: definition of vector space). Occasionally, in very specific circumstances, we may find it useful to draw vectors as lines from the origin, but in those cases vectors are really just n-tuples of real numbers (points in Rⁿ). A plane through the origin is not a vector (not in your situation, anyway), it is a subspace of Rⁿ containing vectors.

EDIT: If you're more "abstract inclined"...

A vector space is just a collection of things (call the collection V) paired with a structure called a field (just think of the real numbers for now). Essentially, we define rules that allow you to add vectors together and multiply them by elements of the field (called scalars). The vectors are elements of the set V.

The vectors in V can be almost anything. In your case, they're n-tuples of real numbers, but in others they could be sets, or planes, or lines, of all sorts of different structures.

 

[chiro]

Not quite understanding what creates a plane, and what creates a line in 3D. Are vectors points in the plane with a line drawn from the origin to the point? Are position vectors vectors drawn between two vectors? If a plane is a vector and a point, how is that different from the description of a line? Not grasping the concepts here...

For one a vector has a finite magnitude whereas a line (or a plane) does not. A ray though is different.

For things that denote position, they are themselves vectors. So are differences in positions, they can also be classified as vectors. Most of this distinction is in the terminology and the context that is being used. A point is a vector but we contextually treat it as a point and not a vector mostly because that in many contexts vectors are treated as 'arrows' with magnitude and direction and while a point has this as well, it may be called a point due to convention and how it is treated in the context that it is being used.

The difference between a plane and a line has to do more with its parametrization than anything else. A line in any dimension can be parametrized with one variable even if its in more than two or three dimensions. A plane can be characterized with two independent variables also regardless if it is in more than three dimensions.

To understand this you need to understand for linear (think straight objects) A line is defined by L = tp + (1-t)q where q and p are points on the line (but not the same point) whereas the definition for a plane is (r - r0) . n = 0 where r0 is a point on the plane and n is a normal vector (preferrably unit normal but this doesn't need to be the case) for the plane. In the line the parameter is t which is a real number and you can get a similar parametrization for the plane if you transform it.

 

[lonewolf219]

Chiro, awesome! Thanks for the descriptions... I definitely understand what you are saying (just an undergrad here!). And number nine, I will read more about vector space...

Thanks guys!