Vector algebra and applications

Question

Can anyone explain the collinearity and coplanarity of vectors ? I'm stuck with the relation between linear dependence, coplanarity, collinearity.

Answer 1

Given a collection of vectors $\{v_1,v_2,\dots,v_n\}$ the dimension of the space spanned by those vectors is the minimal number of vectors needed to form a basis for that space. One common technique in finding the dimension or finding the basis (at least for finite-dimensional vector spaces) is row reduction.

A collection of vectors $\{v_1,v_2,\dots,v_n\}$ are said to be colinear if the space they span is one-dimensional (or less).

A collection of vectors $\{v_1,v_2,\dots,v_n\}$ are said to be coplanar if the space they span is two-dimensional (or less).

Note that any collection of three or more vectors must be linearly dependent if they were to also be colinear or coplanar, but being colinear or coplanar is a stronger more specific condition than simply being dependent. A collection of two vectors is always coplanar regardless of linear independence.