Vectors in a Plane and  Space
     Vectors in a Plane
      Vectors - introduction
         Length, magnitude or norm of the vector
      Collinear, opposite and coplanar vectors
      Addition of vectors
         Triangle rule (law) and parallelogram rule
         Zero or null vector
      Subtraction of vectors
      Scalar multiplication or multiplication of a vector by scalar
         Unit vector
      Addition, subtraction and scalar multiplication of vectors, examples
      Linear combination of vectors
      Linear dependence of vectors
Vectors - introduction
There are physical quantities like force, velocity, acceleration and others that are not fully determined by their 
numerical data. 
For example, a numerical value of speed of motion, or electric or magnetic field strength, not give us the information about direction it move or direction they act.
Such quantities, which are completely specified by a magnitude and a direction, are called vectors or vector
quantities and are represented by directed line segment.
Thus, a vector is denoted as  where the point
A is called the tail or start and point B, the head or tip.
The length or magnitude or norm of the vector a or   is 
Therefore, the length of the arrow represents the vector's magnitude, while the direction in which the arrow points, represents the vector's direction.
A vector with no magnitude, i.e., if the tail and the head coincide, is called the zero or null vector denoted
Collinear, opposite and coplanar vectors
Two vectors are said to be equal if they have the same magnitude and direction or if by parallel shift or translation one could be brought into coincidence with the other, tail to tail and head to head.
Vectors are said to be collinear if they lye on the same line or on parallel lines.
 
Vectors, in the above figure are collinear.
Two collinear vectors of the same magnitudes but opposite directions are said to be opposite vectors.
A vector that is opposite to is denoted as shows the above right figure.
Three or more vectors are said to be coplanar if they lie on the same plane. If two of three vectors are 
collinear then these vectors are coplanar.
To prove this statement, take vectors, of which are collinear.
By using translation bring the tails of all three vectors at the same point. Then, the common line of vectors,
and the line in which lies the vector  determine the unique plane.
Therefore, if vectors are parallel to a given plane, then they are coplanar.
Addition of vectors
The sum of vectors, can be obtained graphically by placing the tail of  to the tip or head of
using translation. Then, draw an arrow from the initial point (tail) of to the endpoint (tip) of to obtain 
the result.
The parallelogram in the above figure shows the addition where, to the tip of by translation, placed
is the tail of then, drawn is the resultant  by joining the tail of to the tip of
Note that the tips of the resultant and the second summand should coincide.
Thus, in the above figure shown is, the triangle rule (law) and the parallelogram rule for finding the resultant or the addition of the two given vectors. The result is the same vector, that is
Therefore, vector addition is commutative.
Since vectors, form a triangle, they lie on the same plane, meaning they are coplanar.
 
Addition of three vectors, is defined as and represented graphically
The above diagrams show that vector addition is associative, that is
The same way defined is the sum of four vectors.
 If by adding vectors obtained is a closed polygon,  then the sum is a null vector.
By adding a vector to its opposite vector , graphically it leads back to the initial point, therefore
so, the result is the null vector.
Subtraction of vectors
Subtraction of two vectors, is defined as addition of vectors that is,
As shows the right figure, subtraction of two vectors can be accomplished directly. 
By using translation place tails of both vectors at the same  point and connect their tips.
Note that the arrow (tip) of the difference coincides with the tip of the first vector ( minuend).
Scalar multiplication or multiplication of a vector by scalar
Scalar is a quantity which is fully expressed by its magnitude or size like length, time, mass, etc. as any real number.
By multiplying a vector a by a real number l obtained is the vector l a  collinear to a but,
                  l times longer then  a  if  | l | > 1,   or    shorter then  a  if  | l | < 1, and
                  directed as  a  if  l > 0,    or    opposite to  if  l < 0,
as is shown in the below figure.
Thus, the magnitude of the vector l a equals to the product of the absolute value of the real number l and the magnitude of the vector a, that is
Besides, for the multiplication of a vector by a real number following rules hold:
 1)   l ( a + b ) = l a + l b
 2)   ( l + m ) a  = l a + m a,   l, m R
 3)   l ( m a )  = m ( l a ) = ( m l ) a
 4)   1 a  =  a,     -1 a  = - a
 5)   0 a  =  0,       m 0  = 0
In the similar triangles ABC and ADE in the right 
figure,         AE : AC = DE : BC = AD : AB = l
therefore,    AE = l   AC.
Since         AE = l a + l b    and    AC = a + b
then,          l a + l b = l ( a + b ).
Unit vector
A vector is called the unit vector of a vector  if
Therefore,
the unit vector determines the direction of the vector
Addition, subtraction and scalar multiplication of vectors, examples
Example:  Given are vectors, determine
Solution:
Example:   Given is a regular hexagon ABCDEF with the center O. Express vectors  CD, BE, EA, and     CE,  in terms of vectors,  AB = a  and  BC = b
Solution:
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 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.
Pre-calculus contents I
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