Lara starts from the school, which is 5 miles west and 7 miles north of the house. she travels 20 miles south, and then 15 miles east. what is her final position? what single translation vector moves her from her starting position to her final position? a truck starts from a warehouse, which is 8 miles south and 3 miles west of the town center. the truck travels 3 miles north, makes a right turn, and then travels for 4 miles. what is the truck’s final position? what single translation vector moves the truck from its starting position to its final position

Problem 1

Let the house be located at (0,0) which is the origin. 

If Lara travels from home to school, then she goes 5 miles west and 7 miles north of her house. 
The school is located at the point (-5,7). Recall that "west" is to the left so we go into the negative x region.

From the school, she travels 20 miles south and 15 miles east.
We will add 15 to the x coordinate so she travels 15 miles east. We go from (-5,7) to (10,7) after adding 15 to the x coordinate. Then subtract 20 from the y coordinate to move 20 miles south. We go from (10,7) to (10,-13)

To recap:
Lara's house is at (0,0)
The school is at (-5,7)
After school she moves to the final point (10,-13)

If we wanted to know what is the fastest way from her house (0,0) to the final point (10,-13), then we would say "move 10 miles east and 13 miles south". Following this path is a direct straight line path. It is the shortest route.

The translation vector would be <x,y> = <15, -20> which says "move 15 units to the right, move 20 units down" if Lara starts at the school and ends up at the final position mentioned earlier.

Your teacher may want you to use the notation (x,y) --> (x+15, y-20) which says the same basic thing.



Problem 2

Point A = (0,0) is the town center
Point B = (-3,-8) is the warehouse location
Point C = (1,-5) is the truck's final position

The same idea will apply as used in problem 1. We start at (0,0) and move 3 miles west and 8 miles south to get to (-3,-8) which is the warehouse location. 

Then we travel 3 miles north to get to (-3,-5) and then 4 miles east landing us on (1,-5) 

The translation vector will be <x,y> = <4,3> because of the statement "truck travels 3 miles north [then east] for 4 miles"