12-5 Geometric Transformations

Quick Review

A change made to a figure is a transformation. The original figure is the preimage, and the transformed figure is the image. A translation slides a figure without changing its size or shape. A dilation changes the size of a figure. You can use matrix addition to translate a figure and scalar multiplication to dilate a figure.

You can use multiplication by the appropriate matrix to perform transformations that are specific reflections or rotations. For example, to reflect a figure across the y-axis, multiply by [ − 1 0 0 1 ] . . matrix with 2 rows and 2 columns , row1 column 1 , negative 1 , column 2 0 , row2 column 1 , 0 , column 2 1 , end matrix . .

Example

A triangle has vertices A(3, 2), B ( 1 , − 2 ) , open 1 comma negative 2 close comma  and C(1, 2). What are the coordinates after a 90° rotation?

[ 0 − 1 1 0 ] [ 3 1 1 2 − 2 2 ] = [ − 2 2 − 2 3 1 1 ] . matrix with 2 rows and 2 columns , row1 column 1 , 0 , column 2 negative 1 , row2 column 1 , 1 , column 2 0 , end matrix . matrix with 2 rows and 3 columns , row1 column 1 , 3 , column 2 1 , column 3 1 , row2 column 1 , 2 , column 2 negative 2 , column 3 2 , end matrix . equals . matrix with 2 rows and 3 columns , row1 column 1 , negative 2 , column 2 2 , column 3 negative 2 , row2 column 1 , 3 , column 2 1 , column 3 1 , end matrix

The coordinates are ( − 2 , 3 ) , ( open negative 2 comma 3 close comma open 2, 1), and ( − 2 , 1 ) . open negative 2 comma 1 close .

Exercises

In matrix form, write the coordinates of each image of the triangle with vertices A(3, 1), B ( − 2 , 0 ) , open negative 2 comma 0 close comma  and C(1, 5).

27. a translation 3 units left and 4 units up
28. a reflection across the y-axis
29. a reflection across the line y = x y equals x
30. a dilation half the original size
31. a dilation twice the original size
32. a rotation of 270 ° 270 degrees

12-6 Vectors

Quick Review

A vector has both magnitude and direction. It is a directed line segment that you can describe using a pair of initial and terminal points. If a vector were in standard position with the initial point at (0, 0), the component form would be (a, b) and the magnitude | v | = a 2 + b 2 absolute value of v , end absolute value , , equals . square root of eh squared , plus , b squared end root  would give you the length.

Given two vectors v = 〈 v 1 , v 2 v sub 1 , comma , v sub 2 〉 and w = 〈 w 1 , w 2 w sub 1 , comma , w sub 2 〉, the dot product v · w v middle dot w  is v 1 w 1 + v 2 w 2 . v sub 1 , w sub 1 , plus , v sub 2 , w sub 2 , .  If the dot product equals 0, then v and w are normal, or perpendicular, to each other.

Example

Are the vectors 〈 − 1 , 2 〉 left pointing angle bracket negative 1 comma 2 right pointing angle bracket  and 〈4, 2〉 normal?

〈 − 1 , 2 〉 · 〈 4 , 2 〉 = ( − 1 ) ( 4 ) + ( 2 ) ( 2 )   = − 4 + 4 = 0 table with 2 rows and 2 columns , row1 column 1 , left pointing angle bracket negative 1 comma 2 right pointing angle bracket middle dot left pointing angle bracket 4 comma 2 right pointing angle bracket , column 2 equals open negative 1 close open 4 close plus open 2 close open 2 close , row2 column 1 , , column 2 equals negative 4 plus 4 equals 0 , end table

The vectors are normal.

Exercises

Let u = 〈 − 3 , negative 3 comma  4〉, v = 〈2, 4〉, and w = 〈4, − 1 negative 1 〉. Write each resulting vector in component form and find the magnitude.

33. u + v u plus v
34. w − u w minus u
35. 3u
36. − 2 w + 3 v negative 2 w plus 3 v
37. 1 2 v + 3 u 1 half , v plus 3 u

38. − w + 3 v + 2 u negative w plus 3 v plus 2 u

    Find the dot product of each pair of vectors and determine whether they are normal.

39. 〈4, − 3 negative 3 〉 and 〈 − 3 , − 4 negative 3 comma negative 4 〉
40. [ 1 7 ] , matrix with 2 rows and 1 column , row1 column 1 , 1 , row2 column 1 , 7 , end matrix  and [ 14 − 2 ] . matrix with 2 rows and 1 column , row1 column 1 , 14 , row2 column 1 , negative 2 , end matrix

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