Friday, December 22, 2006

Geometry 7.4 - Translations and Vectors

I) Translations sec. 7.4

A) Definition: A translation is a transformation that shifts every two points the same distance in the same direction. It is an isometry.

example: Line segment PQ maps to line segment P’Q’.

PQ = P’Q’ , PP’ is parallel to QQ’, PQ is parallel to P’Q’

PP’ = QQ’
Therefore, it creates a parallelogram.

B) Theorem 7.5: If 2 lines are parallel, then a reflection in a line followed by a reflection in another line results in a translation.

EX. line k and line m and parallel and a distance "d" apart.

When you reflect line segment PQ over line k and then over line m, you have P”Q”.

PP” = 2d, PQ mapped to P”Q” is a translation.

PP” is perpendicular to line k, and PP” is perpendicular to line m.

A vector is a quantity that has both direction and magnitude, or size, and is represented by an arrow drawn between two points.

The initial point, or starting point, of the vector is P and the terminal point, or ending point, is Q. The vector name is PQ with the notation of a ray except it does not have the bottom part of the arrow.

The horizontal component of vector PQ is the horizontal shift from point P to point Q and the vertical component of vector PQ is the vertical shift from point P to point Q.

Example, given point P (3,2) and point Q(7,8).

The horizontal shift is from3 to 7 or 4 units.

The vertical shift is from 2 to 8 or 6 units.

the component form is written like the following:

Therefore, for this example, <4,6>.

Translations in a Coordinate Plane:
Sketch a parallelogram with the following vertices:

R(-4, -1), S(-2, 0), T(-1, 3), and U(-3, 2)
Then sketch the image of the parallelogram after translation (x, y) ⇒ (x + 4, y - 2)

R(-4 , -1) ⇒ R’ ( -4 + 4, -1 - 2) = R’ (0 , -3)
S (-2, 0) ⇒ S’ (-2 + 4, 0 - 2) = S’ (2, -2)
T (-1 , 3) ⇒ T’ (-1 + 4, 3 - 2) = T’ (3 , 1)
U (-3 , 2) ⇒ U’ (-3 + 4, 2 - 2) = U’ (1, 0)

The Component Form of this parallelogram would be
<4,-2>.

Draw the vectors between parallelogram RSTU to R’S’T’U’. You have now made a 3-D figure.
To find the magnitude, use the distance formula:
vector's magnitude =
(x1-x2)2+(y1-y2)2

Since the difference of the x’s is the horizontal component
and
the difference of the y’s is the vertical exponent we have:

vector's magnitude =
(horizontal component2 + vertical component2)

So this would be
(42 + (-2)2) = (16 + 4) = 20.