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Find Angles Between Objects in Space with GeoGebra

You can find the angles between two lines in 3D space, a line and a plane and two planes in 3D space by using GeoGebra. You can also read the entry about drawing lines and planes in 3D space.

GeoGebra Instruction 1

Finding the Angle Between Two Lines

1.
Open Algebra View and 3D Graphics under GeoGebra icon View in GeoGebra icon Menu.
2.
In Algebra View, type in the expressions of the two lines that are crossing each other in 3D space. It would be wise to make sure they have a point of intersection, so you know for certain that they will cross each other.
3.
In Algebra View, type

Angle(<line>, <line>)

where you enter the name of each line in each of the <line> fields.

Example 1

Find the angle between the lines a and b with the parametrizations

a : (1, 2, 3) + t (2, 1, 4) b : (1, 2, 3) + t (−4, 5, 9)

First, you type the expressions a and b into Algebra View. If you wish to have an illustration of the lines, you can open 3D Graphics under GeoGebra icon View in GeoGebra icon Menu. Then use

Angle(<line>, <line>)

to find the angle between a and b.

GeoGebra Instruction 2

Finding the Angle Between a Line and a Plane

1.
Open Algebra View and 3D Graphics under GeoGebra icon View in GeoGebra icon Menu.
2.
In Algebra View type in the expressions of your line and plane that are crossing each other. It would be wise to make sure they have a point of intersection, so that they will cross each other.
3.
In Algebra View type in

Angle(<line>, <plane>)

where you type in the name of the line in the <line> field and the name of the plane in the <plane> field.

Example 2

Find the angle between the line a and the plane β, where

a : (1, 2, 3) + t[2, 1, 4] β : y + z − 5 = 0

You type in the expressions of a and β in Algebra View. If you wish to have an illustration, you can open 3D Graphics under GeoGebra icon View in GeoGebra icon Menu. Then use

Angle(<Line>, <Plane>)

to find the angle between the line a and the plane β.

GeoGebra Instruction 3

Finding the Angle Between Two Planes

1.
Open Algebra View and 3D Graphics under GeoGebra icon View in GeoGebra icon Menu.
2.
In Algebra View type in the expressions for two intersecting place. Make sure they actually intersect.
3.
In Algebra View type in Angle(<plane>, <plane>) where you type in the names of the planes in the <plane> fields.

Example 3

Find the angle between the plane α and the plane β given by

α : −13x + 14y + 7z = −36 = 0 β : y + z − 5 = 0

You type in the expressions of α and β in Algebra View. If you wish to have an illustration, you can open 3D Graphics under GeoGebra icon View in GeoGebra icon Menu. Then use

Angle(<plane>, <plane>)

to find the angle between the planes α and β.

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