Vectors in a Plane and  Space
      Vectors and a coordinate system, Cartesian vectors
      Vectors in a coordinate plane (a two-dimensional system of coordinates), Cartesian vectors
         Radius vector or position vector
         Vector components
         Vectors in a two-dimensional system examples
Vectors and a coordinate system, Cartesian vectors
Vectors in a coordinate plane (a two-dimensional system of coordinates), Cartesian vectors
By introducing a coordinate system in a plane with the unit vectors, i and j  (in direction of x and y coordinate axis, respectively) whose tails are in the origin O, then each point of the plane determines a vector rOP.
A directed line segment from the origin to a point P (x, y) in plane is called a radius vector and denoted r.
The radius (or position) vector equals the sum of its vector components, xi and y j  in direction of coordinate axes, that is
   
Consider a vector a in the plane directed from a point P1(x1, y1) to P2(x2, y2) shown in the right diagram, it also equals to the sum of corresponding vector components axi and ayj, in direction of coordinate axes,
   
The diagram shows that are the radius vectors of points P1 and P2,
thus the vector a equals to the difference (joins their heads), that is
 
It is obvious from the above diagrams that a vector and its components, i.e., its projections in direction of 
coordinate axes, form a right triangle, from which, according to Pythagorasí theorem, we determine the length of the vector,
   
Example:   Determine a vector a whose tail is at the point P1(-4, 1) and head at the point P2(-1, -3).
Solution:  Points, P1 and P2 determine radius vectors,
therefore,
Example:   At what point P1(x1, y1) has the vector  a = -7i + 2j  its tail, if its head is at the point 
P2(3, -4)?
Solution: Using or after substitution
Knowing that two vectors are equal if their corresponding scalar (numeric) components are equal, it follows
Example:   To a parallelogram given are vertices, A(-2, 3), B(4, -2) and D(3, 5). Determine coordinates of the vertex C and the intersection point S of the diagonals.
Solution:  To the given vertices point radius vectors,
According to the diagram, the radius vector of the point C,
Hence, the intersection point S of the diagonals S(7/2, 3/2).
Example:   Given are vectors,  a = -2i + 3j  and  b = 4i + a j, determine the coefficient a such that the vectors to be collinear.
Solution:  In order vectors to be collinear must be
as two vectors are equal if their corresponding scalar components are equal, then
Vectors,  a = -2i + 3j  and  b = 4i - 6 j  are collinear, as shows diagram in the above figure.
Vectors in 2D and 3D Contents
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