GeoGebra Quick Start SM025_Vektor

SM025

GeoGebra Classic 6.0

Quick Start
Basic tips for a simpler way in
teaching Mathematics

Kang Kooi Wei
Nurul Syazwani bt Omar

Chow Choon Wooi

What is GeoGebra Classic?

The “GeoGebra Classic” is a free dynamic mathematics app for all levels of education that
brings together geometry, algebra, spreadsheets, graphing, statistics and calculus in one
easy-to-use package. Interactive learning, teaching and evaluation resources created with
GeoGebra can be shared and used by everyone at tube.geogebra.org.

GeoGebra Classic features

 Geometry, Algebra and Spreadsheet are connected and fully dynamic
 Computer Algebra System for symbolic calculations
 3D Graphics View available
 Easy-to-use interface, yet many powerful features
 Authoring tool to create interactive learning materials as web pages
 Available in many languages for millions of users around the world
 Open source software freely available for non-commercial users

Ready to get started?
We hope you enjoy working in GeoGebra Classic!

Sincerely,
G-Maths
The Quick Start Team

G-MATHS 1

Vectors

* Turn on <3D Graphics View> for vector in 3D
Points and vectors may enter via<Input Bar> in Cartesian. Points can also be
created using Point tools and vectors can be created using Vector from Point
Tool or the Vector Tools.

Point
Tools

Vector

Vector
from
Point

Upper case label denoted points, whereas lower case labels refer to vectors.

Example:

Task Commands
(Examples)
a) To enter a point P in 2D in Cartesian form = ( , )
b) To enter a point Q in 3D in Cartesian form = ( , , )
c) To enter a vector u in 2D = ( , )
d) To enter a vector v in 3D = ( , , )
e) To enter a point R in 2D polar form = ( ; °)
f) To enter a point S in 3D spherical coordinates = ( ; °; °)

G-MATHS 2

Notes:

1. Use semicolon (;) to separate polar coordinates. If the command without the

symbol of degree, GeoGebra will treat the angle as radians.
2. Coordinates of points and vectors can be accessed using predefine functions

(), () ().
3. Magnitude of a point Q can be obtained using ( ).
4. Midpoint M of any two points and can be obtained using = ( + )/2.
5. Length of a vector using ℎ( ).

Examples:

If = (3,4), = (7,6) and = (2,4,7).

a) ( ) returns the value of x for .  ( ) = 3

b) ( ) returns the value of y for .  ( ) = 4

c) ( ) returns the value of z for P.  ( ) = 7

d) ( ) returns the magnitude of .  ( ) = 5

e) = ( + )/2 returns the midpoint of P and Q. (5,5)

f) ℎ( ) returns the length of vector  8.31

G-MATHS 3

Basic Operations 4
Magnitude and Unit Vector

a) Abs(<x>)
Gives the magnitude of a vector.
Example
Mathematics: | + + |
: = ( , , )
( )
: .

b) Length(<Object>)
Gives the magnitude of a vector.
Example

Mathematics: | + + |
: = ( , , )

( )
: .

G-MATHS

c) UnitVector(<Object>)

Yields a vector with length1, which has the same direction and orientation as the

given vector. The vector must be defined first.

Example

Mathematics: = + +
: = ( , , )

( )
.
: ( . )
.

Addition/Subtraction of vectors (+/-)
Example

Mathematics: = + + = + + .
( + ) ( − ).

: = ( , , )
= ( , , )
+
−


: + ⟹ ( )

−

− ⟹ (− )
−

Scalar Product (Dot)

a) Dot(<Vector 1>, <Vector 2>)

Returns the value of dot product (scalar product) of two vectors.

Example

Mathematics: ( + + ). ( + + )
: = ( , , )

G-MATHS 5

= ( , , )
( , )

:

Vector Product(Cross, ⊗)
a) Cross(<Vector 1>, <Vector 2>)
Calculates the cross product of two vectors.
Example

Mathematics: ( + + ) ( + + )

: = ( , , )

= ( , , )

( , )

: ( )
−

G-MATHS 6

Note: 7
<Alt><Shift><8> to get the symbol of vector product, ⊗.

Example:
If = + 2 + 3 and = 4 + 5 + 6 , find .

: = ( , , )
= ( , , )
⊗

−
: ( )

−
or

: { , , } ⊗ { , , }
: {− , , − }

Line
a) = + ∗
Mathematics: = ( + + ) + ( + + ))
: ( , , ) + ∗ ( , , )

b) Line(<Point 1>, <Point 2>)
Creates a line through two points A and B
Example
Mathematics: Find the equation of a line passing through
( , , ) ( , , ).
: (( , , ), ( , , ))
: = ( , , ) + ( , , )

G-MATHS

c) Line(<Point>, <Parallel Line>)
Creates a line through the given point parallel to the given line.
Example
Mathematics: Find the equation of a line passing through
( , , ) ( , , ).
: (( , , ), ( , , ))
: Same as above

d) Line(<Point>, <Direction Vector>)

Creates a line through the given point with direction vector v.
Example

Mathematics: Find the equation of a line passing through ( , , ) with the
direction vector = + + .

:
= ( , , )

G-MATHS 8

= ( , , )
( , )
:

e) Curve(<Expression>,<Expression>,<Parameter Variable>,<Start Value>, <End Value>)

Yields the 3D Cartesian parametric curve for the given x-expression, y-expression and z-
expression within the given interval.

Example

Mathematics: Plot the equation of line:
= +
= −
= +

:
( + , − , + , , − , )

G-MATHS 9

:

G-MATHS 10

Plane
a) + + =
Plot a plane.
Example
Mathematics: Plot the plane with the equation of + + =
: + + =
:

b) Plane(<Point>,<Plane>)
Creates a plane through the given point, parallel to the given plane.
Example
Mathematics: Find the equation of the plane passes through ( , , ) and
parallel to the plane + + = .
: (( , , ), ( + + = ))
: + + =

G-MATHS 11

c) Plane(<Point>,<Line>)

Creates the plane through the given point and line.

Example

Mathematics: Find the equation of the plane passes through ( , , ) and
line = ( , , ) + ( , , )

: = ( , , )
: = ( , , ) + ∗ ( , , )
( , )

: − + − =

d) Plane(<Line>,<Line>) 12
Creates the plane through the lines (if the lines are in the same plane).
Example

Mathematics: Find the equation of the plane which contains the
line = (− + + ) + ( + − ) and
line = ( + ) + (− − + )

: : = (− , , ) + ∗ ( , , − )
: = ( , , ) + ∗ (− , − , )
( , )

: − − − = −

G-MATHS

e) Plane(<Point>,<Point>,<Point>)
Creates a plane through three points.
Example

Mathematics: Find the equation of the plane which contains the points
( , , ), ( , , ) ( , , ).

: = ( , , )
= ( , , )
= ( , )
( , , )

: − − + =

G-MATHS 13

f) Plane(<Point>,<Vector>,<Vector>)

Creates the plane through the given point and vectors.

Example

Mathematics: Find the equation of the plane which contains the points
( , , ), = + = + + .

: = ( , , )
= ( , )
= ( , , )
( , , )

: − + =

g) PerpendicularVector(<Plane>)

Creates a perpendicular vector to a plane.

Example

Mathematics: Determine the normal vector of the plane 3x+2y-4z=6.
: : + − =

( )

: ( )
−

G-MATHS 14

G-MATHS 15

Intersection between Line and Plane

a) Intersect(<Object>,<Object>)

Yields the intersection points of two objects.

Example

Mathematics: Find the intersection point between a line = + ( + +
) and a plane + − = .

: : + − =
: = ( , , ) + ∗ ( , , )
( , )

: ( , , )

G-MATHS 16

Angles
a) Angle(<Vector 1>, <Vector 2>)

Returns the angle between two vectors depending on the default angle unit.

Mathematics: Find the angle between = 2 + 2 + 4 and = 5 + 6 + 7 .
:

(( , , ), ( , , ))
: { . °}

b) Angle (<Line 1>, <Line 2>)

Returns the angle between direction vectors of two lines depending on the default angle
unit.

Mathematics:
Line A passes through the points ( , , ) and ( , , ) and line B
connecting points ( , , ) and ( , , ). Find the angle between the
line A and line B.

:
( (( , , ), ( , , )), (( , , ), ( , , )))

: { . °}

c) Angle(<Line>, <Plane>)

Returns the angle between the line and the plane.

Mathematics:
Find the angle between the line = ( ) + ( + + ) and the plane +
− = .
:
= ( , , ) + ∗ ( , , )
: + − =
( , )
: { . °}

G-MATHS 17

d) Angle(<Plane>, <Plane>)
Returns the angle between planes.
Mathematics:
Find the angle between the plane + − = and the plane + − =
.
:
( + − = , + − = )
: { . °}

References

1. https://www.geogebra.org/m/XUv5mXTm#material/seuyvj5u
2. Piecewise functions https://www.youtube.com/watch?v=cg5DVoYEDxE

G-MATHS 18

Appendix

No. Mathematics GeoGebra Keyboard
Notation Command

1+ +
-
2− *
/
3 +−
<Alt>-<>>
4÷ <Alt>-<<>

5±

6≥ >=

7≤ <=

8≠ /=

9 <Alt>-<A>
<Alt>-<B>
10 <Alt>-<G>
<Alt>-<T>
11 <Alt>-<L>
<Alt>-<M>
12 <Alt>-<P>
<Alt>-<U>
13 <Alt>-<E>
<Alt>-<O>
14 ^
<Alt>-<R><x>
15

16 ∞

17

18 °
19

20 √ ( )
21 | | ( )

22 log(a, b)

23 + +

24 −1 ( ) ^ − 1
∗
25 . ( , ) < >< >< ℎ >< 8 >< >

26 ( , )

G-MATHS 19