1. Honors Geometry: Unit 9 Vectors, Matrices, Reflection, Translations, Rotations, Compositions, Clock & Buried Treasure and Dilations Gagan Mavi & Haripriya Suryadevara

2. Vectors Definition: any quantity that has both length and direction. Initial point: where the vector begins. Terminal point: where the vector ends. Magnitude: corresponds to the distance from initial to terminal point. Component form: lists the horizontal and vertical change from initial to terminal point. Common mistakes: forget to write the negative symbol Connections: vectors are similar to rays and they show transformations. Real-life usage: a boat moving to see which direction it goes in.

3. Matrices Definition: Rectangular array of terms, called elements, arranged in rows(m) and columns(n). Adding and Subtracting: Matrices of same dimensions can only be +/- [2*2] and [2*2] Multiplying: Matrices can only be multiplied if # of columns of the 1st = # of rows of 2nd Using Coordinates: (a,b) (c,d) (e,f) Ex:(a*b) (b*c) Common Mistake People usually perform the wrong sign or they don’t know that it’s “NOT POSSIBLE”!! a c e b d f

4. Rotations Definitions: “turn” of an object around a fixed point Key Rules: switch the sign of “y” as you bring it in place of “x” 0°= (x,y) 90°= (-y,x) 180°=(-x,-y) Example: P(-6,-3) Q(-4,-3) R(-2,-5) S(-7,-6) 270°= (y,-x) P’(3,-6) Q’(3,-4) R’(5,-2) S’(6,-7) 360°= (x,y) Matrix multiplication: Parent matrix: 1 0 0 1

5. Reflections ❖ mirror image of a figure Reflect across x-axis: (x,y) (x,-y) ❖ same size and shape Reflect across y-axis: (x,y) (-x,y) Reflect across y=x: (x,y) (y,x) Reflect across y=-x: (x,y) (-y,-x) Common mistake: mix up coordinate rule and use the wrong matrix. Example: Using matrix multiplication… A’ B’ C’ -2 3 4 -5 3 -6

6. Translations Definition: A “slide” of a figure from one position to another. ❖ every point moves the same distance and direction ❖ written as slide,, right _, up _, (x+_,y+_) Common mistakes: move image in the wrong direction. Example:

7. Composition ❖ Glide Reflection; rotations, translations, and reflections combined Reflect y=x; (0,2) (1,5) (2,2) (2,0) (5,1) (2,2) Translate; (0,2) (1,5) (2,2) (3,5) (6,6) (3,7) Translate;(5,2)(6,5)(4,7)(2,5)(3,2) : (-3,2)(-2,5)(-4,7)(-6,5)(-5,2) Reflection;(5,2)(6,5)(4,7)(2,5)(3,2):(-3,-2)(-2,-5)(-4,-7)(-6,-5)(-5,-2) Common Mistake People mix up the order of the transformations and the answer gets messed up. Connection Relate to surface area and volume when more than one shape is combined to make a composition shape.

8. Clock & Buried Treasure Start at 5:00 1. Reflect over x-axis 2. Rotate 60° CCW 3. Reflect across y-axis 4. Rotate 120° CW Result: 7:00 Buried Treasure Perform the same steps as the clock problems but in a grid. Common Mistake Some people may miss one step, confuse the x and y axis or confuse CCW or CW.

9. Dilations ❖ stretch or shrink an object ❖ reduce or enlarge shape by scale factor ❖ reduction if 0<k<1 ❖ enlargement if k>1 Common mistake: Dilate by the wrong number or enlarge instead of reduction Connection: Have to know how to multiply and divide. Related to similar shapes since all similar shapes are dilations. Real-life usage: When using a scale model for a building. Example:

10. You need to Geomet-try hard for this test! Good luck everyone, we are so close! Credit to Mrs. Street for the joke!! Thanks Mrs. Street for helping us through the year. We will miss you!!!