ACTIVITY 2.7
VECTORS AND TRANSLATIONS

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 +
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.

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

END OF ACTIVITY 2.7
VECTORS AND TRANSLATIONS