An introduction to vectors is available here. Let's make a step further and look at when vectors are collinear and coplanar, and what is the basis of a plane and space.
The vectors 
and 
are collinear if they are parallel or lie on parallel line beams.
If the vectors and are collinear and is nonzero, then there is a 
that holds:
Three vectors are coplanar if they lie in the same plane.
If the vectors 
and 
are coplanar, then there exist 
such that:
A linear combination of vectors is a simple sum of several different vectors multiplied by any real numbers.
The linear combination of the vectors 
and 
is each expression written as the sum of these two vectors multiplied by any real numbers k and l:
The result of a linear combination are different vectors, depending on the given vectors and the values of real numbers. For real numbers k = l = 0 and given vectors and we get as a result a zero vector:
Such vectors, in which their linear combination is a zero vector only in the case when 
, are called linearly independent otherwise vectors are linearly dependent. Intuitive: vectors are linearly dependent if they are parallel, otherwise they are linearly independent.
The vectors and are linearly independent if they can for any k and l their sum
add to 0 if and only if 
and 
. Otherwise, the vectors are linearly dependent.
The vectors and are linearly dependent if and only if they are collinear.
The linear combination of the vectors 
and 
is each expression written as the sum of these three vectors multiplied by any real numbers k, l and m:
The result of a linear combination are different vectors, depending on the given vectors and the values of real numbers. For real numbers k = l = m = 0 and given vectors and we get a zero vector as a result:
Such vectors, in which their linear combination is a zero vector only when k = l = m = 0, are called linearly independent, otherwise the vectors are linearly dependent. Intuitive: vectors are linearly dependent if they are coplanar, otherwise they are linearly independent.
The vectors and are linearly independent if for any k, l and m their sum
add to 0 if and only if they are simultaneously k = 0, l = 0 and m = 0. Otherwise, the vectors are linearly dependent.
The vectors and are linearly dependent if and only if they are coplanar.
Now let's look at:
What are non-parallel vectors? We say that they are linearly independent or non-collinear and
What are the vectors that do not lie in the same plane? They are linearly independent or non-coplanar.
The base is orthogonal if the base vectors are perpendicular to each other; normalized if the vectors are uniform (length one) and orthonormal if the vectors are uniform and perpendicular to each other.
