Unit 9 ReviewBy: 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 TranslationsFind the coordinates of A’B’C’ after a translation by the rule (x,y) (x+5, y-3).

To solve this, you would performthe 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 ProblemStart: 2Rotate 90 degrees ccwReflect over x-axisEnd: ?

Answer to Example of a Clock Problem

AnswerStart: 2Rotate 90 degrees ccw: 11Reflect over x-axisEnd: 7

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

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

Answer to Buried Treasure Example

Start: (1,1)Translate: right 3, down 4Reflect over y-axisTranslate: left 2, up 1End: (-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 ExampleFind 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.AnswerLengths: (2 x 2) = 4Widths: (3 x 3) = 9

Common mistakes: Accidentally dividing bythe scale factor instead of multiplying them.