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
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 lie 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 than  a  if  | l | > 1,   or    shorter than  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 ).
Vectors in 2D and 3D Contents
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