what does it mean to say a vector lies in a plane in R^3. Does it mean it's inital point lies on the plane?

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- Oct 10th 2011, 11:58 AMDukevector lies in the plane
what does it mean to say a vector lies in a plane in R^3. Does it mean it's inital point lies on the plane?

- Oct 10th 2011, 12:07 PMDevenoRe: vector lies in the plane
usually it means that the vector can be written as a linear combination of two other vectors you know beforehand.

one vector determines a line, two vectors determine a plane.

there are two different ways of thinking of a vector: one is simply as a point.

another way, is as an "arrow" going from one point in space, to another. in this view, all "arrows" with the same length and direction are considered "equal".

the bridge between these two views is to identify a point in space with the arrow that goes from the origin, to that point. - Oct 10th 2011, 12:18 PMDukeRe: vector lies in the plane
I was thinking how to visualize it.

- Oct 10th 2011, 12:25 PMPlatoRe: vector lies in the plane
The wording of this question is problematic.

In one sense all vectors have the same initial point: $\displaystyle (0,0,0).$

Vectors are really just equivalence classes, defined by length and direction. Thus as reply #2 suggests that any two vectors can be though of as being in some plane, though that may not be a unique plane.

It is true that any three non-colinear points determine a unique plane. - Oct 10th 2011, 12:32 PMDevenoRe: vector lies in the plane
the plane is unique if you also say it must pass through (0,0,0) (which goes along with thinking of all vectors having intial point (0,0,0)).

if one regards each vector as "unique", then one has the counter-intuitive result that vectors do not form a vector space (vector addition not being defined

unless the tail of one is at the head of another). put more technically, an affine plane isn't quite the same thing as a linear plane.

in ordinary low-dimensions that most people are familiar with: the line ax+b is not a linear function. - Oct 10th 2011, 12:38 PMPlatoRe: vector lies in the plane
- Oct 10th 2011, 12:58 PMDevenoRe: vector lies in the plane
valid point. the vectors must be linearly independent, or we have a "degenerate plane" (a line).