Question

If we have any two vectors in space, can we always find a plane that contains them? Please explain.

Answer 1

It is possible when they are subspace of a same vector space .

Suppose U and V are subspace of a vector space W . Also suppose that { u₁ , u₂ , .....,u_n } forms a basis of U and { v₁ , v₂ , v₃ , ,,,,v_m } be a basis of V .

Then , Span { u₁ , u₂ ,.....,u_n , v₁ , v₂ , v_{3 ....,}v_m } is a plane containing both of U and V .

But if U and V are subspace of different vector space then it is not possible to find a plane containing both of them .

For example let ,

U = Span { ( 1 , 0 ) } then U is a subspace of

V = Span { ( 1 , 0 , 0 ) } then V is a subspace of

Then it is not possible to find a plane containg both of them .

.

.

.

If you have any doubt or need more clarification at any step please comment .