Vectors in a Plane and  Space
Vectors in a Plane
Addition, subtraction and scalar multiplication of vectors, examples
Linear combination of vectors
Linear dependence of vectors
Addition, subtraction and scalar multiplication of vectors, examples
Example:  Determine the distance of the midpoint M, of the segment P1P2, and the point O, if points, P1 and P2 are heads of vectors p1 and p2 respectively, and whose tails coincide with the point O as shows the  figure.
 Solution: The vector P1P2 represents the difference p2 - p1.
 Example:  In a triangle ABC drown are medians as vectors,
 Prove that
 Solution:   Replacing the sides of the triangle by vectors, directed as in the diagram,
Example:   Use vectors to prove that line segments joining the midpoints of adjacent sides of a quadrangle,
form a parallelogram.
Solution:  The sides of the quadrangle ABCD are replaced by vectors, directed as in the diagram so that the
opposite vectors connecting midpoints are,
As equal vectors are of same lengths and parallel, therefore the line segments connecting the midpoints of any quadrangle, form a parallelogram.
Example:   Determine a vector which coincides with the angle bisectors of vectors, a and b in the diagram.
Solution:  The unit vectors of the given vectors are,
 and they form a rhombus whose diagonal is
 That is, a vector will coincide with
 the angle bisector, while a vector
defines the angle bisector of the supplementary angle of vectors, a and b.
Linear combination of vectors
A vector  dl a + m blm Î R  denotes the linear combination of the vectors, shown in the left diagram.
On the same way, the vector  e = l a + m b + n c  represents the linear combination of the vectors, a, b and c, as shows the right diagram in the above figure.
Example:   Given is a parallelogram ABCD, the midpoint M of the side AD and the intersection point O of the diagonals. Express vectors, AO, BD, and BM  as the linear combination of vectors, aAB and b = AD.
 Solution:
Example:   To a cuboid ABCDEFGH the point M divides the edge EH in the ratio EM : MH = 2 : 1 and the point N the edge AB in the ratio AN : NB = 3 : 2. Express the vector MN as the linear combination of vectors, a = AB, bAD and cAE.
 Solution:
 MN is the linear combination of the vectors, a, b and c.
Linear dependence of vectors
Vectors, a = OA and bOB whose points, O, A and B all lie on the same line are said to be linear dependent, but if the points, O, A and B do not all lie on the same line then a and b, are not collinear, and are said to be linear independent.
a)  If a and b are linear independent vectors then every vector d of the plane determined by a and b, can be written as the linear combination of these vectors, that is in the form
Vectors, a = OA, bOB and cOC, whose points, O, A, B and C all lie on the same plane, are said to be coplanar or linear dependent. But if points, O, A, B and C do not all lie on the same plane, then a, b and c are not coplanar, and are said to be linear independent.
b)  If, a, b and c are linear independent vectors in 3D space then every vector d, from the space, can be
represented as
that is, as the linear combination of the vectors, a, b and c.
Thus, in the last example the sides of the cuboid are replaced by three linear independent vectors, a, b and c so that the vector MN can be written as the linear combination of the vectors, a, b and c.