# Points and Vectors

#### GeoGebra Objects

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 and vectors can be created using the Vector from Point Tool or the 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.
Examples:
• 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 Spreadsheet View, name it using its cell address, e.g.: `A2 = (1, 0)`
Notes:
• 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).
Examples:
• 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°.

## 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)` 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.

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