Points and Vectors

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Points and vectors may be entered via Input Bar in Cartesian or polar coordinates (see Numbers and Angles). Points can also be created using Mode point.svg Point tools and vectors can be created using the Mode vectorfrompoint.svg Vector from Point Tool or the Mode vector.svg Vector Tool and a variety of commands.

Note: Upper case labels denote points, whereas lower case labels refer to vectors. This convention is not mandatory.
  • To enter a point P or a vector v in 2D in Cartesian coordinates you may use P = (1, 0) or v = (0, 5).
  • To enter a point P or a vector v in 3D in Cartesian coordinates you may use P = (1, 0, 2) or v = (0, 5, -1).
  • To enter a point P in 2D in polar coordinates, you may use P = (1; 0°) or v = (5; 90°).
  • To enter a point P in 3D in spherical coordinates, enter three coordinates of the type (ρ, θ, φ) like e.g. P = (1; 60°; 30°).
  • To enter a point in the Menu view spreadsheet.svg Spreadsheet View, name it using its cell address, e.g.: A2 = (1, 0)
  • You need to use a semicolon to separate polar coordinates. If you don’t type the degree symbol, GeoGebra will treat the angle as if entered in radians
  • Coordinates of points and vectors can be accessed using predefined functions x() and y() (and z() for 3D points).
  • Polar coordinates of point Q can be obtained using abs(Q) and arg(Q) (and also alt(Q) for 3D points).
  • If P=(1,2) is a point and v=(3,4) is a vector, x(P) returns 1 and y(v) returns 4.
  • abs(P) returns 2.24 and arg(P) returns 26.57°.


In GeoGebra, you can also do calculations with points and vectors.

  • You can create the midpoint M of two points A and B by entering M = (A + B) / 2 into the Input Bar.
  • You may calculate the length of a vector v using length = sqrt(v * v) or length = Length(v)
  • You can get the coordinates of the starting and terminal point of a vector v using the commands Point(v, 0) and Point(v, 1) respectively.
  • If A = (a, b), then A + 1 returns (a + 1, b + 1). If A is a complex number a+bί, then A+1 returns a + 1 + bί.

Vector Product

Let (a, b) and (c, d) be two points or vectors. Then (a, b) ⊗ (c, d) returns the z-coordinate of vector product (a, b, 0) ⊗ (c, d, 0) as single number.

Similar syntax is valid for lists, but the result in such case is a list.

  • {1, 2} ⊗ {4, 5} returns {0, 0, -3}
  • {1, 2, 3} ⊗ {4, 5, 6} returns {3, 6, -3}.
© 2020 International GeoGebra Institute