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Download Ch 16 Geometric Transformations and Vectors Combined Version 2
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Transcript
Ch 16 Geometric Transformations Learning Goal Learn to transform geometric figures using matrix operations Vocabulary preimage original figure image new/transformed figure Types of Transformations translations rotations dilations reflections Image Preimag e Translations shift preimage left/right/up/down use matrix addition Translate the figure with the following coordinates 8 units to the right and 5 units down. 5,5 , 3, 7 , 1,5 , 3,1 Dilations Enlarge/reduce the size of a figure use scalar multiplication Double the size of the preimage. 1 3 2 0 2 1 Rotating a Figure Rotation turns a figure about a fixed point uses matrix multiplication Rotation Matrix • Preimage = Image Rotation Matrices for the Coordinate Plane (counterclockwise) 90 180 270 360 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 Reflecting a Figure Reflection maps a point or figure in the coordinate plane to its mirror image using a specific line as its line of reflection uses matrix multiplication across x-axis 1 0 0 1 Reflection Matrix • Preimage = Image Reflection Matrices for the Coordinate Plane across y-axis 1 0 0 1 across y x 0 1 1 0 across y - x 0 1 1 0 Ch 16 Vectors Goal: Learn to use basic vector operations and the dot product What is a Vector? A mathematical object that has both magnitude & direction Directed line segment w/initial and terminal points Note: 2 vectors w/same magnitude & direction are considered to be equivalent vectors, no matter their locations Representing a Vector Graphically Component Form (w/initial pt @ origin) Matrix Form Magnitude Matrix transformations that can be applied to vectors Translations Rotations Dilations Reflections Rotating a Vector Write vector in matrix form rotation matrix vector resulting vector ◦ Ex 1. Rotate the vector w = <3,-2> by 90 . What is the component form of the resulting vector? Operations with Vectors Givens: v = <v1 , v2 > and w = <w1 , w2 > and any real number k addition v w v1 w1 , v2 w2 subtraction (not commutative) v w v1 w1 , v2 w2 scalar multiplication k>0 and ≠ 1 changes the magnitude k<0 and ≠ -1 changes the magnitude & reverses direction kv kv1 , kv2 dot product (scalar value; not a vector) helps determine if 2 vectors are normal (normal only if dot product = 0) v w v1w1 v2 w2 Are the following vectors normal (perpendicular)? (Show Proof) Ex 2 t 2, 5 and u 7,3 Ex 3 v 10, 4 and w 2,5 Ex 4: Application A twin engine airplane travels south at a speed of 300 mi/h in still air. The plane encounters a wind blowing 50 mi/h due east. What is the resultant speed of the airplane? (Use the tip-to-tail method)
