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Parallel translation. Vectors
Let us take two parallel straight lines \latex{t_1} and \latex{t_2} with a distance of d between them and triangle ABC. Let us reflect this triangle about straight line \latex{t_1}, then let us reflect the image triangle A’B’C’ about straight line \latex{t_2}. (Figure 62)
Figure 62
\latex{ A }
\latex{ B }
\latex{ l }
\latex{ C' }
\latex{ A' }
\latex{ B' }
\latex{ B}
\latex{ C}
\latex{ A}
\latex{ y }
\latex{ z }
\latex{ x }
\latex{ x }
\latex{ x' }
\latex{ x' }
\latex{ z' }
\latex{ y' }
\latex{ y' }
\latex{ y }
\latex{ z' }
\latex{ z }
\latex{ C }
\latex{ l }
When considering the relative positions of triangle A’’B’’C’’ resulting after the second reflection and of the original triangle ABC and using the properties of the line reflection we can realise the following:
- \latex{AA’’ = BB’’ = CC’’ = 2d};
- Straight lines AA’’, BB’’, CC’’ are parallel with each other and are perpendicular to the axes.
Doing the above mentioned reflections with parallel axes in succession can be replaced by a transformation which moves every single point of the plane in a direction perpendicular to the axes with the same distance (the double of the distance of the axes). This transformation is the parallel translation.
The properties of parallel translation
- If the axes of the two line reflections do not coincide, then the parallel translation does not have a fixed point. If the axes coincide, then we get the identical transformation, where every point of the plane is a fixed point.
- The straight lines parallel with the direction of translation (perpendicular to the axes) are invariant straight lines. If the parallel translation is not an identical transformation, then there is no other invariant figure.
- A straight line is parallel with its translated image. (Figure 63)
- Parallel translation is a distance-preserving and angle-preserving transformation.
- Parallel translation can be produced by doing two line reflections about parallel axes in succession where the distance of the translation is double the distance of the axes and its direction depends on the order of reflections. (It can be proven that: if the distance and the direction of a parallel translation is given, then one of the axes of the defining line reflections can be chosen arbitrarily out of those which are perpendicular to the direction of the translation, but this choice unambiguously defines the other axis.)
- Since doing two line reflections in succession does not change orientation, parallel translation is an orientation-preserving transformation.
the properties of parallel translation
Figure 63
\latex{ e' }
\latex{ e}
\latex{ e}
\latex{ l_{1} }
\latex{ l_{2} }
Example 1
Let us take triangle ABC, and let us translate it parallel with its side AB by double the length of side AB.
Solution
The exercise has two possible solutions, since the direction in which we translate the triangle is not fixed. By measuring the length of side AB on the straight line of side AB from A and from B we get one vertex of each of the two possible triangles: \latex{A'_1} and \latex{B'_2}. By using the properties of parallel translation it is now easy to construct the two image triangles. (Figure 64)
Figure 64
\latex{ A'_{2} }
\latex{ B'_{2} }
\latex{ C'_{2} }
\latex{A }
\latex{B }
\latex{C }
\latex{C'_{1} }
\latex{A'_{1} }
\latex{B'_{1} }
The concept of a vector
The parallel translation can be given by doing two line reflections with parallel axes in succession, but it can also be given by the direction and the distance of the translation simultaneously. The latter two characteristics are united in a very important concept of mathematics, in the vector. Henceforth we are going to have a closer look at this concept.
The parallel translation from the introductory example can unambiguously be defined by giving line segment AA’’ and by distinguishing its two end-points: A is the starting point, A’’ is the end-point. With this we have given the directed line segment pointing from A to A’’. The directed line segment pointing from B to B’’ has the same length, points in the same direction and gives the same parallel translation as the one pointing from A to A’’.
DEFINITION: A directed line segment unambiguously defines a vector.
DEFINITION: Two vectors are equal if they define the same parallel translation.
If the starting point of the directed line segment is A, its end-point is B, then the notation of the vector defined by it is: \latex{\overrightarrow{AB}} (Pronounced: “ vector AB”.) (Figure 65)
Figure 65
\latex{ A }
\latex{ B }
Vectors are usually also denoted by a single lowercase letter, e.g. \latex{\overrightarrow{a}, \overrightarrow{b}}(Figure 66). In printed text “bold” lowercase letters are also used to denote vectors: \latex{a, b}.
Figure 66
\latex{\overrightarrow a}
\latex{\overrightarrow b}
DEFINITION: The length of the directed line segment defining the vector is called the absolute value or magnitude of the vector.
Notation: \latex{\left|\overrightarrow{AB}\right|}, \latex{\left|\overrightarrow{a}\right|}, \latex{\left|a\right|}
Notation: \latex{\left|\overrightarrow{AB}\right|}, \latex{\left|\overrightarrow{a}\right|}, \latex{\left|a\right|}
DEFINITION: Two vectors are parallel if the straight lines of the directed line segments defining them are parallel. (Figure 67)
Figure 67
\latex{ A }
\latex{ B }
\latex{ f }
\latex{ E }
\latex{ F }
\latex{ D }
\latex{ C }
\latex{ g }
It can be seen that two vectors can also be parallel so that they are pointing in the opposite direction. Before defining the unidirectionality of two vectors we define the unidirectionality of the defining directed line segments.
\latex{AC\cap BD=\varnothing}
Figure 69
\latex{ A }
\latex{ C }
\latex{ B }
\latex{ D }
DEFINITION:
- Two non-collinear directed line segments are unidirectional (in the same direction) if their straight lines are parallel and the two line segments connecting the two starting points and the two end-points have no interior point in common. (Figure 68)
- Two collinear directed line segments are unidirectional (in the same direction) if there is a directed line segment coinciding a different straight line with which both of them are unidirectional. (Figure 69)
Figure 68
\latex{ E }
\latex{ F }
\latex{ A }
\latex{ B }
\latex{ C }
\latex{ D }
DEFINITION: Two vectors are unidirectional if the directed line segments defining them are unidirectional.
DEFINITION: Two vectors are in the opposite direction if they are parallel but not unidirectional. (Figure 70)
DEFINITION: If two vectors have the same magnitude (absolute value) and are in the opposite direction, then the two vectors are each other's opposite. Notation: the opposite of \latex{\overrightarrow{a}} is –\latex{\overrightarrow{a}}. (Figure 71)
DEFINITION: Two vectors are in the opposite direction if they are parallel but not unidirectional. (Figure 70)
DEFINITION: If two vectors have the same magnitude (absolute value) and are in the opposite direction, then the two vectors are each other's opposite. Notation: the opposite of \latex{\overrightarrow{a}} is –\latex{\overrightarrow{a}}. (Figure 71)
Figure 70
\latex{ A }
\latex{ B }
\latex{ C }
\latex{ D }
DEFINITION: Two vectors are equal if they are unidirectional and their magnitudes are equal.
Unidirectional directed line segments of equal length define the same vector.
Figure 71
\latex{\overrightarrow a}
-\latex{\overrightarrow a}
Notes:
- Graphically we represent vectors by directed line segments, just as we have done so far, and instead of the expression directed line segment we mostly use the word vector. Based on the above it can be seen that several unidirectional directed line segments of equal length can represent the same vector, thus we have the opportunity to choose the one that suits the given problem the best. (Figure 72)
- In our studies we have come across mathematical objects, which although had the same meaning, still appeared different. Let us think of the rational numbers, the fractions. A fraction can arbitrarily be expanded, the numerator and the denominator of the expanded form are different from the numerator and the denominator of the original fraction, but they still denote the same number. For example:
\latex{\frac{1}{2}=\frac{2}{4}=\frac{3}{6}=\dots=\frac{625}{1250}=\dots.}
In this respect the concept of vector is similar to the concept of fraction: a given vector can be represented by infinitely many directed line segments just as a given fraction has infinitely many forms.
Figure 72
\latex{\overrightarrow a}
\latex{\overrightarrow a}
\latex{\overrightarrow a}
Example 2
The vertices of the cube in figure 73 define the following vectors:
\latex{\overrightarrow{AB}}; \latex{\overrightarrow{BC}}; \latex{\overrightarrow{BE}}; \latex{\overrightarrow{BF}}; \latex{\overrightarrow{CG}}; \latex{\overrightarrow{CH}}; \latex{\overrightarrow{HG}}; \latex{\overrightarrow{FE}}; \latex{\overrightarrow{HE}}
Which vectors are equal to each other? Which vectors are each other's opposites?
Figure 73
\latex{ E }
\latex{ H }
\latex{ G }
\latex{ F }
\latex{ C }
\latex{ B }
\latex{ D }
\latex{ A }
Solution
\latex{\overrightarrow{AB}=\overrightarrow{HG}}, \latex{\;\overrightarrow{BF}=\overrightarrow{CG}}, \latex{\;\overrightarrow{BE}=\overrightarrow{CH}}.
\latex{\overrightarrow{AB}} and \latex{\overrightarrow{FE}}, \latex{\overrightarrow{HG}} and \latex{\overrightarrow{FE}}, \latex{\overrightarrow{BC}} and \latex{\overrightarrow{HE}} are opposite vectors.
\latex{\overrightarrow{AB}} and \latex{\overrightarrow{FE}}, \latex{\overrightarrow{HG}} and \latex{\overrightarrow{FE}}, \latex{\overrightarrow{BC}} and \latex{\overrightarrow{HE}} are opposite vectors.
Example 3
Let us take triangle ABC and point P. Let us translate the triangle by \latex{\overrightarrow{AP}}.
Solution
The image of point A is \latex{P = A’}, and \latex{\overrightarrow{AP}=\overrightarrow{BB'}=\overrightarrow{CC'}} So we can construct the vertices of the image triangle so that we mark off distance AP on the straight lines parallel with straight line AP and passing through B and C in the suitable direction. (Figure 74)
Figure 74
\latex{ C }
\latex{ C' }
\latex{ B }
\latex{ A }
\latex{ B' }
\latex{ P=A' }
Exercises
{{exercise_number}}. The interior angle bisector starting from the right-angled vertex of right-angled triangle ABC intersects hypotenuse AB at point D. Let us translate the triangle by \latex{\overrightarrow{CD}}.
{{exercise_number}}. From the planar figures in the figure select those pairs which are each other's image when translated parallel.
\latex{ A }
\latex{ B }
\latex{ C }
\latex{ D }
\latex{ E }
\latex{ F }
{{exercise_number}}. Take a circle with \latex{ 2 } \latex{ cm } long radius and chord \latex{ AB } inside. Let us translate the circle by \latex{\overrightarrow{AB}}.
{{exercise_number}}. Take a triangle and two straight lines. Translate the triangle parallel with one of the straight lines so that
- its centroid;
- the centre of its inscribed circle
lies on the other straight line. Is there a case when the exercise cannot be solved?
{{exercise_number}}. Decide which of the following statements are true and which are false.
- There is a point set in the plane which is transformed into itself when translated parallel.
- Doing two line reflections in succession results in a parallel translation.
- The parallel translation transforms every straight line into a straight line parallel with itself.
- The parallel translation can have a fixed point.
- The parallel translation preserves the orientation of the figures.
{{exercise_number}}. Take a \latex{45^{\circ}} angle and a line segment outside the angular domain. Translate the line segment so that each of its end-points lies on one arm.
- the common part of the original triangle and of the image triangle;
- the union of the original triangle and the image triangle?
{{exercise_number}}. Which of the vectors in the figure are equal and which are each other's opposites?
\latex{\overrightarrow a}
\latex{\overrightarrow b}
\latex{\overrightarrow c}
\latex{\overrightarrow d}
\latex{\overrightarrow e}
\latex{\overrightarrow f}
\latex{\overrightarrow g}
\latex{\overrightarrow h}
\latex{\overrightarrow i}
\latex{\overrightarrow j}
{{exercise_number}}. Towns A and B are separated by a river which has the same width everywhere. A public road is planned to be built between the two towns so that the bridge arching over the river is perpendicular to the river and the length of the road connecting town A with town B is the shortest possible. Construct the shortest road based on the figure.
\latex{ A }
\latex{ B }