Sequences of Translations 04:01 minutes
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Transcript Sequences of Translations
The sun is coming up, as this family of moles is getting ready for bed. Doug, the youngest, likes a bedtime story before he falls asleep. His favorite tale is about the bright sun and its journey across the sky. Doug wants to see the sun from his favorite tale so he makes a plan to sneak to the surface. In order to find his way there, he needs to use a sequence of translations. Here's a diagram of Doug's underground neighborhood. We'll represent Doug with this polygon, 'D'. In order to translate polygon 'D', we need vector, 'u'. Vector 'u' is a directed line segment that tells us how far Doug will move, and in what direction. Let's translate polygon 'D' along vector 'u'. The image that results from this translation is called 'D' prime. Doug makes excellent progress towards the surface but this stone is an obstacle. We'll need a SECOND translation along a new vector to avoid the stone. Let's translate 'D' prime along vector 'v'. Translating D prime along v we see the resulting image let's call it 'D' double-prime. The sequence of translations along these two vectors brings Doug to the surface. Can you see a single translation, that would have brought Doug to the surface in one step? Translating 'D' along this vector gives the same result as translating along 'u', and 'v'. We'll call this vector 'AB'. When 'u' and 'v' are arranged as shown, 'AB' has the same starting point as 'u' and the same endpoint as 'v'. Ah, the sun is warm! But it's so bright and hard to see! Ok, maybe it's time to go. Doug is outta here! How can we help Doug return home quickly? Vector 'AB' gives us the correct distance, but it's the wrong direction. If we want the same distance as 'AB' but the opposite direction we can translate along 'BA', which means it starts here and ends here. We need to translate 'D' double-prime along vector 'BA'. This translation maps D double-prime back onto the original 'D'. We applied three translations to D. First, we translated 'D' along 'u' then along 'v' and then along 'BA'. This sequence of translations mapped 'D' onto itself. Doug ended his day where he started. Before we conclude his tale, let's review what we learned. When we translate a figure along a vector then translate its image along another vector this is called a sequence of translations. The figure is not changed by this process, only moved. We can always find a single vector that accomplishes the same sequence of translations in one step. We can also reverse the direction of vectors. Reversing vectors can help us create sequences of translations that map the original figure back onto itself. Doug is back home but his family is nowhere to be found! Oh no, they used their own sequence of translations to find their way to the surface, in search of Doug! These moles really dig translations! But their timing is a little off!