One of the main applications of the vectors are the translations.
Make a translation of a point P according to a vector consists in moving the point P until point P' so that = .
Observe that if = (v 1, v 2), P (p 1, p 2) and P' (p'1, p'2), then
P' = P +![](gif/vec_v.gif)
or, in components: (p'1,p'2) = (p 1, p 2) + (v 1, v 2)
That is to say, to obtain P' there is enough with to add to P the components of .
COMPOSITION OF TRANSLATIONS
To apply from consecutive form to one P (p 1, p 2) point two or more translations given by the vectors = (a 1, a2), = (b 1, b 2), = (c 1, c 2), ... it is added to P coordinates the components of , , , ...
P '= P + + + + · · ·
or in components (p'1, p'2) = (p1, p2) + (a1, a2) + (b1, b2) + (c1, c2) + ···
Because the vector sum is commutative, the composition of translations also will be commutative. |
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INTERACTIVE ACTIVITY
This construction represents the three vector sum + + , but also P can represent one composition of three translations applied to the point P successively that brings it to the point P'. You take advantage of this construction to make the following compositions of three translations applied to the point P graphically:
1) Three given translations by
= (4,3), = (-2,2) and = (8,1)
2) Three given translations by
= (-3,4), = (5,-1) and = (9,0)
3) Three given translations by
= (0,-4), = (4,8) and = (6,-6)
4) Three given translations by
= (2,2), = (3,3) and = (4,4)
SOLUTION |
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You make in your work notebook the following compositions of three translation applied to any point:
a) Three given translations by = (5,7), = (6,-5) and = (-11, -2.)
b) Given by = (-5, -2), = (10,0) and = (0,10.)
c) Given by = (9,9), = (0,-9) and = (-12,8.)
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