A vector can be thought of as an arrow with a direction
and a length. Vectors are often used to translate
(move) objects. In this sketch, the vector translates the
red polygon to the blue polygon; that is, the direction
of the vector gives the direction to move, and the length of the vector gives the distance
to move.
If you have Geometer's Sketchpad installed on your computer,
you can get the original sketch.
Move the head (arrow end) of the vector.
Watch how the blue polygon moves.
Move the tail of the vector.
Watch how the blue polygon moves.
Move the tail of the vector to a point on the red polygon. Notice how
the head is at the corresponding point on the blue polygon.
One way to describe a vector is to give its length and direction (for example, as an angle
from the positive x axis). Another way is to give the x and y coordinates of
its tail and head. The most common way is to agree that the tail must be at (0,0), and give only the x and y coordinates of the head.
Group/lab problems
In the interactive diagram, move the tail of the vector to (0,0) and the
head of the vector to (0,1). Pick a point on the red polygon and find its
coordinates. Where is the corresponding point on the blue polygon?
Now move the tail of the vector to (2,1) and the head to (2,2).
How do the direction and length compare to those of the vector in the previous item?
Look at the same point on the red polygon and answer the same question.
Do this for several more vectors: for the pairs of vectors in the table below, compare their directions and lengths and their effect on
the polygon.
First vector tail
First vector head
Second vector tail
Second vector head
(0,0)
(2,0)
(-1,1))
(1,1)
(0,0)
(2,1)
(-3,-2))
(-1,-1)
(0,0)
(-1,1)
(-3,2))
(-4,3)
What is the pattern for the pairs of vectors chosen in these questions?
How are the head and tail coordinates for the second vector related to those for the
first vector?
Formulate your discoveries into a general rule for relating a vector whose
tail is not at (0,0) to one whose tail is at (0,0).
Individual homework problems
In the following questions, use the convention for naming a vector where
the tail is at the origin; then the coordinates of the head are the coordinates of the vector.
Give the coordinates of the vector (call it v1) that translates polygon A to polygon B1.
Give the coordinates of the vector (call it v2) that translates polygon B1 to polygon C.
Give the coordinates of the vector (call it v3) that translates polygon A to polygon B2.
Give the coordinates of the vector (call it v4) that translates polygon B2 to polygon C.
Give the coordinates of the vector (call it v5) that translates polygon A to polygon C.
Find a relation between the coordinates for v1, v2, and v5.
Explain the relationship in words and/or symbols.
Your relationship should also work for v3, v4, and v5. Does it?
Using graph paper, draw vector v1 as an arrow starting at a point on
polygon A. Draw vector v2 starting at the endpoint of v1.
Draw vector v5 as an arrow starting at the same point on
polygon A. Describe in words the geometric relationship you get.
Your relationship should also work for v3, v4, and v5. Does it? Draw the corresponding diagram.
Give the coordinates of the vector v6 that translates polygon D to polygon E.
Give the coordinates of the vector v7 that translates polygon E to polygon D.
The vector v6 moves D to E, and v7 moves E back to D.
So v7 "undoes" v6; that is, the vectors are "inverses".
Find a relationship between the coordinates of inverse vectors.
Using graph paper, draw vector v6 as an arrow starting at a point on
polygon D. Draw vector v7 starting at the endpoint of v6.
Describe in words the geometric relationship you get.
What vector would translate the point (0,0) to (3,2)?
What vector would translate the point (3,2) to (0,0)?
What vector would translate the point (3,2) to (4,3)?
What vector would translate the point (4,3) to (3,2)?
State a general rule for the vector that translates point (a,b) to point (c,d).
Point F has coordinates (1,2). First it is translated by vector (-2,4); then it
is translated by (5,-7); then by (0,2); then by (-3,-3); then by (1,2). Where does it
end up?
