Representation

a vector v can be interpreted as:

direction (x, y, z) and length |v|, arrow, a translation, connection between two points (see Subtrcation), as point (if the vector is a bounded vector to the origin), force, speed, velocity ....

as symbol:
letters, often bold or underlined v, v

as three numbers (x, y, z):
the vvvvector space is build up by combination of three orthogonal 1-dimensional directions, the x-, y- and z-axis. every direction that is not parallel to the axes could be described as a linear combination of three directions (vectors) that are parallel to the axes. a linear combination here is adding three vectors (see Addition). to include the second aspect of a vector (the length) the three (axis) vectors n1, n2, n3 are set to length one, this is a orthonormal base. to build all other vectors with this base, the three base vectors length gets scaled with the numbers x, y, z and then the vectors are added together:

v = xn1 + yn2 + z*n3 =

because the n vectors are (1, 0, 0), (0, 1, 0) and (0, 0, 1) the result is:

v = ( 1x, 0x, 0x ) + ( 0y, 1y, 0y ) + ( 0z, 0z, 1*z ) =

= ( x, 0, 0 ) + ( 0, y, 0 ) + ( 0, 0, z ) = (x, y, z)