1. By: Suhas Navada and Antony Jacob
Unit 9 Review
By: Suhas Navada and Antony Jacob

2. Overview of the unit Vectors Matrices Translations and Reflections
Rotations
Clock problems and Buried Treasure
Dilations

3. Intro to Vectors
Definition: a quantity that has both direction and length
Initial point: where the vector begins
Terminal point: where the vector ends
Component form: lists the horizontal and vertical change from initial point to terminal point
Magnitude: length of a vector. Distance from initial point to terminal point. Can be found with distance formula.
Amplitude: direction of a vector. Angle the vector makes with the positive x-axis. Can be found with SOH-CAH-TOA.
Equal vectors: have same magnitude and direction
Parallel vectors: have same slope
Real-life example: the path that a pool ball takes while playing pool

4. Intro to Vectors cont.
Resultant vector: a vector that represents the sum of two given vectors

5. Matrices Definition: rectangular array of terms called elements
Elements are arranged in rows (m) and columns (n)
Dimensions of a matrix: m x n
Row Matrix: has 1 row
Column Matrix: has only one column
Square Matrix : number of rows must be same as number of columns
Zero Matrix: all elements are zero
Matrix Addition and Subtraction: only matrices of the same dimensions can be added or subtracted
Real-life example: Matrices are used in common surveys

6. Matrices cont.
Multiplying matrices:

7. Example Problem Using the picture subtract C-B How to Solve
The new matrix must have the same dimensions of the first two matrices.
You must subtract corresponding elements of the matrices. (i.e. 4-(-1), 1-0, etc.)

8. Answer Common Mistakes
Switching the order of the matrices while performing the operation.

9. Reflections and Translations
Definition of a reflection: It is the mirror image of a figure produced by “flipping” it over a line of reflection.
A common mistake is that if one axis is given, people mistakenly reflect over the opposite axis.
Definition of a translation: It is a “slide” of a figure from one position to another. Every point on the figure moves in same direction and same distance. Can be described using vectors or a motion rule.

10. Reflections and Translations cont.
Real-life Example for reflections: Mirrors reflect images in a plane
Real-life Example for translations: Pushing a box from one side of the room to another.

11. Example of Translations
Find the coordinates of A’B’C’ after a translation by the rule (x,y) (x+5, y-3).
To solve this, you would perform
the motion rule [eg. (1+5, 3-3)].

12. Answer to Example of Translation
A’ = (x+5, y-3) = (-3+5, 1-3) = (2, -2)
B’ = (x+5, y-3) = (1+5, 3-3) = (6,0)
C’ = (x+5, y-3) = (2+5, -4+3) = (7,-1)

13. Rotations
Definition- a “turn” of an object around a fixed point - The center of rotation
The shape is congruent to the original shape
You can the (X,Y) formulas to find the coordinate of it after the rotation which are found in the Picture below

14. Common Mistakes
One of the most common mistakes is the rules for rotation which are easy to get mixed up.
A real example of rotations are motors which turn a set amount of rotations.

15. Clock Problems
Every number represents 30 degrees . So if the hour hand was in 9 and the minute hand is on 3. So the it would be 180 degrees
Common mistakes are going the wrong direction for rotating if no direction is given (eg. rotate 150 degrees).

16. Example of a Clock Problem
Start: 2
Rotate 90 degrees ccw
Reflect over x-axis
End: ?

17. Answer to Example of a Clock Problem
Start: 2
Rotate 90 degrees ccw: 11
Reflect over x-axis
End: 7

18. Buried Treasure Problems
Buried Treasure Problems are like clock problems except instead of on a clock, the buried treasure problems are done on a coordinate plane.

19. Buried Treasure Example
Start: (1,1)
Translate: right 3, down 4
Reflect over y-axis
Translate: left 2, up 1
End: ?

20. Answer to Buried Treasure Example
Start: (1,1)
Translate: right 3, down 4
Reflect over y-axis
Translate: left 2, up 1
End: (-6,-2)

21. Dilations If the scalar is less than 1 it is a reduction
Scalar multiplication-multiplication of a vector or point by a scalar (k).
If dilation is greater than 1 it is an enlargement
If the center of dilation is in the shape - you find the center of the shape and the distance from the center to the vertices. Then you multiply it by the scalar to get the points of the new shape.
If the center of dilation is on the shapes edges or vertices - you find the distance from the given side point to all the vertices. Then multiply that distance by the scalar then graph the new distances
If the center of dilation is outside the shape - you must draw a line from the outside point. Then you must multiply the distance by the scalar and graph the new points.

22. Dilation Example Find the new dimensions of this shape.
The scale factor is 2.
This can be solved by multiplying the dimensions by the scale factor, which is 2.
Answer
Lengths: (2 x 2) = 4
Widths: (3 x 3) = 9
Common mistakes: Accidentally dividing by
the scale factor instead of multiplying
them.