Points and Vectors

Points and vectors may be entered via Input Bar in Cartesian or polar coordinates (see Numbers and Angles). Points can also be created using Point tools, Vector from Point Tool, Vector between Two Points Tool and a variety of commands.

Note: Upper case labels denote points, whereas lower case labels refer to vectors. This convention is not mandatory.

Example:

• To enter a point P or a vector v in Cartesian coordinates you may use P = (1, 0) or v = (0, 5).
• In order to use polar coordinates type in P = (1; 0°) or v = (5; 90°).

Note: 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.

Example: If P=(1,2) is a point and v=(3,4) is a vector, x(P) returns 1 and y(v) returns 4.

Polar coordinates of point Q can be obtained using abs(Q) and arg(Q).

Calculations

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

Example:

• 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)
• 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

For two points or vectors (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.

Example:

• {1, 2} ⊗ {4, 5} returns {0, 0, -3}
• {1, 2, 3} ⊗ {4, 5, 6} returns {3, 6, -3}.