Vectors in a Plane and  Space
      Vectors in three-dimensional space in terms of Cartesian coordinates
         Vectors in a three-dimensional coordinate system examples
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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