Unit 9: Vectors, Matrices and Transformations By: Sushy Balraj and Sonal Verma

Vectors Any quantity that contains both length and direction. It is named from the initial to the terminal point. Example 1: ⇀ Name: OA Component Form: <2,3> 3 up 2 right Component form shows the horizontal and vertical change of the vector. Naming vectors are just like naming rays (Unit 1). We name both from the endpoint (initial) to the arrow (terminal).

Vectors Common Mistake: People forget to put the brackets around the component form of the vector. Magnitude of a vector relates to the length of it. To find magnitude use the distance formula: Amplitude is the direction in which the arrow points. To find the amplitude use: SOHCAHTOA. We learned how to find unknown side lengths and angle measurements of triangles using sine, cosine, and tangent (Unit 7 Part 2).

Real Life Example of Vectors What is the component form for vector AB and vector BC? AB: <12-0, 4-0> = <12,4> BC: <16-12, 2-4> = <4, -2>

Matrices Rectangular array of elements. Arranged in rows and columns. In order to add or subtract matrices, the dimensions need to be the same. Example 2: (3x3), (3x3)

Matrices Multiplying matrices… Common Mistake: People often forget the prerequisites to add, subtract and multiply the matrices.

Real Life Example of Matrices STEP 2 STEP 1

Examples of Matrices A+B= (2x4) (4x3)

Translations Sliding a figure from one position to another. The shape and size stays congruent to the original figure. Use motion rules, component forms, or vectors to indicate translation. Units move up (+), down (-), left (-), right (+). Common Mistake: People forget to put the negative behind the coordinate may lead to an incorrect translation. (x, y) (x-5, y-2)

Reflections Mirror images flipped over “line of reflection” Every point of reflection is the same distance from the line of reflection as the corresponding point on the original figure If reflecting over... x-axis: (x,y) (x, -y) y-axis: (x,y) (-x, y) y=x: (x,y) (y, x) y=-x: (x,y) (-y, -x)

Rotations Turning an object Direction can be counterclockwise (CCW) or clockwise (CW), if not specified- then always CCW Coordinate Rules- 90 CCW: (x,y) (-y, x) [0 -1] [1 0] 180 CCW: (x,y) (-x, -y) [-1 0] [0 -1] 270 CCW: (x,y) (y, -x) [0 1] [-1 0] 360 CCW: (x,y) (x,y) [1 0] (Parent matrix) [0 1]

Rotation Use matrix multiplication to figure out the points for rotation by multiplying the matrix by the coordinates. Struggle: Memorizing the coordinate rules for rotations. Common Mistake: 90° CW is actually 270° CCW!

Dilations stretch or shrink k= scale factor enlargement= k>1 reduction= 0<k<1 Example 3: Use the given point and k=2 to dilate the figure. Remember to use a ruler 📏to get precise measurements!

Composition of Transformations combining transformations translation + reflection = glide reflection Example 4: The vertices of triangle ABC are A(-2, -2), B(-2, -4), C(-4, -4). List the coordinates after a composition of transformation. Dilation: centered at the origin with a scale factor of 2 Reflection: across the y-axis Answer: Dilation- multiply everything by 2; A(-4,-4), B(-4, -8), C(-8, -8) Reflection over y-axis: (-x,y); A(4, -4), B(4, -8), C(8, -8) A(4,-4) B(4,-8) C(8, -8)

Buried Treasures and Clock Problems Different ways to approach transformations Example 5: Start: 10 Rotate: 180° Reflect: x-axis Rotate: 150°CCW What time is it? Example 6: Start: (6,-1) Translate: (x+2, y-1) Reflect: x-axis Rotate: 180° Component: <-2, 1> Where is the treasure?

Now there’s no possible way you could fail this unit :)