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

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

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

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

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

Matrices cont. Multiplying matrices:

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.)

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

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.

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.

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)].

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)

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

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.

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).

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

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

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.

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

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)

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.

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.