Vectors in a Plane and  Space
      Vectors in three-dimensional space in terms of Cartesian coordinates
         Angles of vectors in relation to coordinate axes, directional cosines - scalar components of a vector
         The unit vector of a vector
         Vectors in a three-dimensional coordinate system, examples
Vectors in three-dimensional space in terms of Cartesian coordinates
By introducing three mutually perpendicular unit vectors, i, j and k, in direction of coordinate axes of the 
three-dimensional coordinate system, called
standard basis vectors, every point P(x, y, z) of the space 
determines the radius vector or the position vector,
  or and   the length of  
Similarly, a vector a in the right diagram, which is directed from a point P1(x1, y1, z1) to a point P2(x2, y2, z2) in space, equals to sum of its vector components,  axi, ay j, and azk,  in the direction of the coordinate axes, x, y, and z respectively, that is
   
The vector a represents the difference of the radius vectors,
thus
   
so the scalar (numeric) components of the vector a,
Therefore, the length of the vector a,
   
Angles of vectors in relation to coordinate axes, directional cosines - scalar components of a vector
The scalar components of a vector and its magnitude form a right triangle in which the hypotenuse equals the magnitude of the vector, then
since  b = 90 - a  then  cosb = sina,
and components of a vector a,
If a vector a in 3D space forms with the coordinate axes, x, y and z angles, a, b and g respectively, then 
components of the vector are,
   
while for the radius vector of the point P(x, y, z),
   
Therefore, for angles, a, b and g hold the expressions,
   
that are called the directional cosines of the vector a.
The unit vector of the vector a,
   
   
This condition must satisfy angles that a vector in three-dimensional space form with the coordinate axes.
For vectors in a coordinate plane and since  b = 90 - a, then 
 cosb = cos(90 - a) = sina,  follows the basic trigonometric identity.
Similarly, the unit vector of a radius vector in space
   
determined by corresponding directional cosines.
Therefore the radius vector forms with coordinate axes angles,
   
As vectors are uniquely determined by its components or coordinates, they are usually denoted using matrix algebra notation, in the coordinate plane,
and in the 3-D space
Vectors in a three-dimensional coordinate system examples
Example:   Determine angles that a radius vector of the point A(3, -2, 5) forms with the coordinate axes. 
Solution:  Let calculate the magnitude or length of the radius vector,
Angles between the radius vector and the coordinate axes are,
Example:   A vector AB is directed from point A(-1, -2, 1) to point B(-2, 3, 4), find the unit vector of the vector AB.
Solution: Determine the vector AB from the expression
The length of the vector AB
The unit vector of the vector AB
Check that the directional cosines of the unit vector satisfy the relation,

Example:  A vector a in a 3D-space, of the length | a | = 4, forms with axes, x and y the same angles, ab = 60, find the components (coordinates) of the vector a.

Solution:  Using relation
applying given conditions,
The components of a ,
Example:  Show that vectors,  a = -i + 3 j + kb = 3i - 4 - 2 and  c = 5i - 10 - 4k are coplanar. 
Solution:  If all three vectors lie on the same plane then there are coefficients, l and m such that, for example c = la + mb, i.e., each of the vectors can be expressed as the linear combination of the remaining two.
Example:  Points, A(0, -2, 1), B(-2, 1, -3) and C(3, -1, 2) are the vertices of a triangle, determine the vector of the median  mc = CM  and its length.
Solution:  The radius vector of the midpoint of the side AB,
The vector of the median CMmc = CM = rm - rc,
and the length of the median CM
To check over the obtained result, calculate the coordinates of the centroid G,
The centroid divides every median in the ratio 2 : 1, counting from the vertex to the midpoint, therefore
Vectors in 2D and 3D Contents
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