Ostukorv on tühi
Geometric objects in space
The world around us, the objects and solids in it perceived through our senses inspired the concepts allowing us to mathematically describe space. Today, these elements created through abstraction react on our environment, creating thousands of different shapes.
Mutual position of geometric objects
The most important concepts in \latex{ 3 }-dimensional geometry are: the point, the line and the plane. These objects are so simple that we do not need to define them thanks to the images connected to them by anyone.
Depending on their position and relation to each other we can distinguish several cases.
For two lines the possibilities are the following (Figure 1):
Depending on their position and relation to each other we can distinguish several cases.
For two lines the possibilities are the following (Figure 1):
Two lines are intersecting if they have exactly one point in common.
Two lines are parallel if they are contained in the same plane and they do not intersect.
Two lines are skew if they are not contained in one plane.
Two lines are parallel if they are contained in the same plane and they do not intersect.
Two lines are skew if they are not contained in one plane.
\latex{e\cap f=M}
\latex{e||f (e\cap f= \varnothing )}
\latex{ e } and \latex{ f } are skew
\latex{ M }
\latex{ f }
\latex{ e }
\latex{ e }
\latex{ f }
\latex{ e }
\latex{ f }
Figure 1
We can examine the relation between two planes similarly (Figure 2):
Two planes are intersecting if they have exactly one line in common.
Two planes are parallel if they do not intersect.
Two planes are parallel if they do not intersect.
\latex{S_{1}\cap S_{2}=e}
\latex{S_{1}||S_{2}}
\latex{ S_2 }
\latex{ S_1 }
\latex{ e }
\latex{ S_1 }
\latex{ S_2 }
Figure 2
A line can either be incident to a plane, in which case every point of it is also a point of the plane, or it can intersect the plane at one point, or they do not have any points in common, in which case the line and the plane are parallel. (Figure 3)
\latex{e\subset S}
\latex{e\cap S=M}
\latex{e||S}
\latex{ e }
\latex{ S }
\latex{ M }
\latex{ S }
\latex{ e }
\latex{ e }
\latex{ S }
Figure 3
For any plane and any point outside the plane there is exactly one plane containing the point that is parallel to the first plane.
A plane can be uniquely described by
- three points which are not collinear;
- two intersecting lines;
- two parallel lines;
- a line and a point not on the line.
A plane divides the space into two half-spaces, in which case the plane is called the bounding plane of the half-spaces.
Example 1
How many parts can \latex{ 3 } planes divide the space into?
Solution
We have to consider the feasible positions of the three planes.
The following cases are possible:
The following cases are possible:
- They can be pairwise parallel, in which case four parts of space are created.
(Figure 4/a) - There can be two parallel with the last one intersecting both, in this case there are six parts. (Figure 4/b)
- If there are no parallel planes, then there are three other possibilities. The number of parts in each case is seven, six and eight. (Figure 5)
\latex{ S_3 }
\latex{ S_2 }
\latex{ S_1 }
\latex{ S_1 }
\latex{ S_2 }
\latex{ S_3 }
\latex{ b })
\latex{ a })
Figure 4
Figure 5
Example 2
How many regions the planes do incident to the faces of a cube divide the space into?
Solution
Let us enumerate the regions by their relative position to the cube. (Figure 6)
There is \latex{ 1 } region inside the cube. There are \latex{ 6 } regions for which a face of the cube is part of its boundary. There are \latex{ 12 } regions which belong to the edges of the cube and \latex{ 8 } which is incident to the vertices of the cube.
We have listed every region, and there are \latex{1+6+12+8=27.}
There is \latex{ 1 } region inside the cube. There are \latex{ 6 } regions for which a face of the cube is part of its boundary. There are \latex{ 12 } regions which belong to the edges of the cube and \latex{ 8 } which is incident to the vertices of the cube.
We have listed every region, and there are \latex{1+6+12+8=27.}
Figure 6
Angle of objects in three dimensions
Two rays with common initial points divide the plane into two parts. We call these parts angles.
We have the same definition for interpreting different angles in \latex{ 3 }-dimensional space, using that the proposition for corresponding angles is true in space as well, that is, if the sides of the angles \latex{ AOB } and \latex{ A’O’B’ } are parallel and they point in the same directions, then the two angles are equal. (Figure 7)
We have the same definition for interpreting different angles in \latex{ 3 }-dimensional space, using that the proposition for corresponding angles is true in space as well, that is, if the sides of the angles \latex{ AOB } and \latex{ A’O’B’ } are parallel and they point in the same directions, then the two angles are equal. (Figure 7)
\latex{\alpha =\alpha '}
\latex{\alpha}
\latex{\alpha'}
\latex{ A }
\latex{ A' }
\latex{ B' }
\latex{ O' }
\latex{ O }
\latex{ B }
Figure 7
the angle between \latex{ e } and \latex{ f= } the angle between \latex{ e' } and \latex{ f' }
\latex{\beta}
\latex{ e }
\latex{ f }
\latex{ f' }
\latex{ e' }
Figure 8
DEFINITION: The angle between a pair of skew lines is the angle formed by two lines parallel to them which intersect in an arbitrary point. (Figure 8)
During the inspection of the angle between a line and a plane we have to be careful about the perpendicular position.
DEFINITION: A line and a plane are perpendicular if the line is perpendicular to every line of the plane.
This statement is hard to verify since we would have to check infinitely many pairs; but the following theorem can be proven.
THEOREM: If a line is perpendicular to both of two intersecting lines of a plane, then it is perpendicular to every line of the plane, hence it is perpendicular to the plane itself.
The orthogonal image of a point onto a plane is the intersection point of the line passing through the point and perpendicular to the plane, and the plane itself (Figure 9). The above described transformation is called orthogonal projection. It is easy to verify that
- the orthogonal image of a line is a line;
- the images of parallel lines are parallel lines (these can coincide or a pair of points can also be the image);
- the ratio of segments on a line is maintained by the projection;
- the image of any point in the plane is itself.
\latex{ P }
\latex{ P' }
\latex{ S }
Figure 9
DEFINITION: If a line intersects a plane but not perpendicular to it, then the inclination of the line and the plane is the angle between the line and its perpendicular image in the plane. (Figure 10)
The inclination of a plane and a line parallel to it is \latex{ 0° }.
The inclination of a plane and a line parallel to it is \latex{ 0° }.
\latex{ S }
\latex{ e }
Figure 10
DEFINITION: When looking for the inclination of two intersecting planes, we take the perpendicular lines in both planes to the line of intersection in an arbitrary point of the line of intersection. The inclination of the two planes is the angle between the two lines described above. (Figure 11)
The inclination of two parallel lines is \latex{ 0° }.
The inclination of two parallel lines is \latex{ 0° }.
Figure 11
Distance of three-dimensional objects
The computation of the distance of two point sets usually goes as follows. We connect every possible pairs of points from the two sets, and the length of the smallest of these segments (if exists) is the distance of the two point sets.
The definitions below are results of the previous method.
The definitions below are results of the previous method.
DEFINITION: The distance of a point and a plane is the distance of the point and its orthogonal image on the plane. (Figure 12/a)
DEFINITION: The distance of a plane and a line parallel to it is the distance of the plane and an abritrary point of the line. (Figure 12/b)
DEFINITION: The distance of a plane and a line parallel to it is the distance of the plane and an abritrary point of the line. (Figure 12/b)
DEFINITION: The distance of two parallel planes is the distance of one plane and an arbitrary point of the other one. (Figure 12/c)
DEFINITION: The distance of two skew lines is the length of the segment which is on the line that perpendicularly intersects both lines bounded by the two lines. This line is called the normal transversal of the two skew lines. (Figure 12/d)
DEFINITION: The distance of two skew lines is the length of the segment which is on the line that perpendicularly intersects both lines bounded by the two lines. This line is called the normal transversal of the two skew lines. (Figure 12/d)
\latex{ P }
\latex{ P' }
\latex{ P' }
\latex{ P' }
\latex{ P }
\latex{ e }
\latex{ e }
\latex{ P }
\latex{ d_{e,\,S } }
\latex{ d_{P,\,S} }
\latex{ d_{S_1,\,S} }
\latex{ f }
\latex{ S }
\latex{ S_1 }
\latex{ S }
\latex{ S }
\latex{ b })
\latex{ a })
\latex{ c })
\latex{ d })
Figure 12
Example 3
The cuboid \latex{ ABCDEFGH } is given. Its edges are: \latex{AB=6\, cm} , \latex{BC= 6\, cm}, \latex{AE=5\, cm} long. (Figure 13). Determine the following distances:
- the distance of vertex \latex{ A } and edge \latex{ GC };
- the distance of vertex \latex{ A } and plane \latex{ BDH };
- the distance of edge \latex{ HG } and plane \latex{ EDC }.
\latex{ G }
\latex{ H }
\latex{ H }
\latex{ G }
\latex{ C }
\latex{ C }
\latex{ T }
\latex{ D }
\latex{ A }
\latex{ B }
\latex{ B }
\latex{ A }
\latex{ D }
\latex{ E }
\latex{ F }
\latex{ F }
\latex{ E }
Figure 13
Solution (a)
The distance in question is the length of the diagonal of rectangle \latex{ ABCD } as\latex{ AC } is perpendicular to the edge\latex{ CG }. Thus
\latex{AC=\sqrt{6^{2}+4^{2} }=\sqrt{52}=2\times \sqrt{13}\approx 7.21cm.}
Solution (b)
The line perpendicular to the plane of the triangle \latex{ BDH } through vertex \latex{ A } coincides with the altitude line corresponding to the hypotenuse \latex{ BD } of the right angled triangle \latex{ ABD }. Let us denote the foot of this altitude line with \latex{ T }.
By expressing the area of the triangle in two different ways the length of the segment \latex{ AT } can be determined:
By expressing the area of the triangle in two different ways the length of the segment \latex{ AT } can be determined:
\latex{\frac{6\times 4}{2}=\frac{2\times \sqrt{13}\times AT }{2},}
\latex{12=\sqrt{13}\times AT,}
\latex{AT=\frac{12}{\sqrt{13} }=\frac{12\times \sqrt{13} }{13}\approx 3.33cm.}
\latex{12=\sqrt{13}\times AT,}
\latex{AT=\frac{12}{\sqrt{13} }=\frac{12\times \sqrt{13} }{13}\approx 3.33cm.}
Solution (c)
The intersection of the plane \latex{ EDC } and the cuboid is the parallelogram \latex{ EDCF }. The line \latex{ HG } and the plane in question are parallel.
To determine their distance, take the line perpendicular to the plane \latex{ EDC } through, for example, the point \latex{ H }. This perpendicular line is actually the line of altitude corresponding to the hypotenuse \latex{ ED } of the right-angled triangle \latex{ EDH }. Let us denote the foot of this altitude by \latex{ V }. Using the formula for the area of a triangle the distance \latex{ HV } can be determined:
To determine their distance, take the line perpendicular to the plane \latex{ EDC } through, for example, the point \latex{ H }. This perpendicular line is actually the line of altitude corresponding to the hypotenuse \latex{ ED } of the right-angled triangle \latex{ EDH }. Let us denote the foot of this altitude by \latex{ V }. Using the formula for the area of a triangle the distance \latex{ HV } can be determined:
\latex{\frac{5\times 4}{2}=\frac{\sqrt{5^{2}+4^{2} }\times HV }{2},}
\latex{20=\sqrt{41}\times HV,}
\latex{HV=\frac{20}{\sqrt{41} }=\frac{20\times \sqrt{41} }{41}\approx 3.12cm.}
\latex{20=\sqrt{41}\times HV,}
\latex{HV=\frac{20}{\sqrt{41} }=\frac{20\times \sqrt{41} }{41}\approx 3.12cm.}
Exercises
{{exercise_number}}. How many regions is the space divided into by the planes of the faces of a tetrahedron?
{{exercise_number}}. How many regions can \latex{ 4 } planes divide the space into if
- at least \latex{ 3 } of them are parallel;
- \latex{ 2 } of them are parallel?
{{exercise_number}}. The cube \latex{ ABCDEFGH } with sides of length a is given. Determine the distance of
- the vertex \latex{ A } and the edge \latex{ GC };
- the vertex \latex{ A } and the plane \latex{ BDH };
- the planes \latex{ BDE } and \latex{ CFH }.
{{exercise_number}}. How far are two non-adjacent edges of the regular tetrahedron with sides of length \latex{ a }? (The regular tetrehedron is a solid bounded by four equilateral triangles.)
{{exercise_number}}. Determine the angle between two skew face diagonals of the cube. (There are multiple cases.)
{{exercise_number}}. Determine the angle between a face diagonal and a space diagonal of the cube. (There are multiple cases.)
{{exercise_number}}. What is the distance of two opposite vertices of a cube with sides of length \latex{ a }? What is the shortest path on the surface of the cube on which a bug can reach a vertex from the opposite one? What is the angle between a segment of this shortest path and a space diagonal? What is the angle between segment of the path and a face diagonal?
{{exercise_number}}. The planes \latex{\alpha} and \latex{\beta} are perpendicular to each other; point \latex{ A } is an arbitrary point of \latex{\alpha,} point \latex{ B } is an arbitrary point of \latex{\beta}. \latex{ A }’ and \latex{ B }’ denote the perpendicular image of point \latex{ A } and \latex{ B }, respectively, on the line of intersection of the planes; and let \latex{ F } denote the midpoint of the segment \latex{ AB }. Prove that \latex{FA'=FB'.}
{{exercise_number}}. Every point of the unit cube is coloured by one of three colours. Is it true that there are always two points with the same colour with distance being at least \latex{ 1.4 }?
Puzzle
Four ships are moving in formation close to each other on the sea: the distance between any two is \latex{ 3\, km }. There is a cruiser, a tanker and a cargo ship among them. What kind of ship is the fourth one?