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Another way to describe a translation is with the use of a vector.
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A
vector is a quantity
that has magnitude (length) and direction.
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| A
vector is represented by a
directed line segment,
which is a segment with an
arrow at one end indicating the direction of movement.
Unlike a ray, a directed line segment has a specific length.
The
direction
is indicated by an arrow pointing from the
tail
(the initial point) to the
head
(the terminal point). If the tail is at point A
and the head is at point B, the vector from
A to B is written as:
notation:
(Vectors may
also be labeled as a single bold face letter, such as vector v.) |
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| The length
(magnitude) of a vector v is written |v|.
Length is always a non-negative real number.
As you can see in the diagram at the right, the
length of a vector can be found by forming a right triangle and
utilizing the Pythagorean Theorem or by using the Distance
Formula.
The vector at the right translates 6 units to
the right and 4 units upward. The magnitude of the vector
is
from the Pythagorean Theorem, or from the
Distance Formula:
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direction of a vector is determined
by the angle it makes with a horizontal line.
In the diagram at the right, to
find the direction of the vector (in degrees) we will utilize
trigonometry. The tangent of the angle formed by the
vector and the horizontal line (the one drawn parallel to the
x-axis) is 4/6 (opposite/adjacent).
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Translations and vectors: The translation at the left
shows a vector translating the top triangle 4 units to the right
and 9 units downward. The notation for such vector movement may
be written as:
or
Vectors such as those used in translations are what is known as
free vectors. Any two vectors
of the same length and parallel to each other are considered
identical. They need not have the same initial and
terminal points. |
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A free vector is an infinite
set of parallel directed line segments and can be thought of as
a translation. Notice that the vectors in this translation
which connect the pre-image vertices to the image vertices are
all parallel and are all the same length.
You may also hear the terms
"displacement" vector or "translation" vector when working with
translations. |
Position vector:
To each free vector (or translation),
there corresponds a position vector
which is the image of the origin
under that translation.
Unlike a free vector, a position vector is
"tied" or "fixed" to the origin. A position vector
describes the spatial position of a point relative to the
origin. |
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