# The coordinates of a point, vector or line in the plane: coordinates

### 13.13.4 The coordinates of a point, vector or line in the plane: coordinates

See Section 14.12.4 for coordinates in three-dimensional geometry.

The coordinates finds the coordinates of a point or two points that determine a line.

- coordinates takes one argument:
  X, either a point, a sequence or list of points, or a line.
- coordinates(X) returns:
  - a list consisting of the abscissa and ordinate of X, if X is a point or a vector, or a sequence or list of such lists, if X is a sequence or list of points.
  - a list of two points on the line X, in the order determined by the direction of the line, if X is a line.

#### Examples.

- Input:
  coordinates(1+2*i)
  or:
  coordinates(point(1+2*i))
  or:
  coordinates(vector(1+2*i))
  Output:

  ⎡ ⎣ | 1,2 | ⎤ ⎦

- Input:
  coordinates(point(1+2*i) - point(i))
  or:
  coordinates(vector(i,1+2*i))
  or:
  coordinates(vector(point(i),point(1+2*i)))
  or:
  coordinates(vector([0,1],[1,2]))
  Output:

  ⎡ ⎣ | 1,1 | ⎤ ⎦

- Input:
  d:= line(-1+i,1+2*i)
  or:
  d:= line(point(-1,1),point(1,2))
  then:
  coordinates(d)
  Output:

  ⎡ ⎣ | −1+i,1+2 i | ⎤ ⎦

- Input:
  coordinates(line(y = (1/2 * x + 3/2)))
  Output:

  ⎡
  ⎢
  ⎢
  ⎣

  3 i
  2

  ,1+2 i

  ⎤
  ⎥
  ⎥
  ⎦

- Input:
  coordinates(line(x - 2*y + 3 = 0))
  Output:

  ⎡
  ⎢
  ⎢
  ⎣

  3 i
  2

  ,

  −4+i
  2

  ⎤
  ⎥
  ⎥
  ⎦

- Input:
  coordinates(i,1+2*i)
  or:
  coordinates(point(i),point(1+2*i))
  Output:

  ⎡ ⎣ | 0,1 | ⎤ ⎦ | , | ⎡ ⎣ | 1,2 | ⎤ ⎦

- Note that if the argument is a list of real numbers, it is interpreted as a list of points on the real axis.
  Input:
  coordinates([1,2])
  Output:

  ⎡
  ⎢
  ⎣

  1 | 0
  2 | 0

  ⎤
  ⎥
  ⎦