Unit 9: Vectors, Matrices and Transformations By: Sushy Balraj and Sonal Verma.
-
Upload
vanessa-richardson -
Category
Documents
-
view
244 -
download
0
Transcript of Unit 9: Vectors, Matrices and Transformations By: Sushy Balraj and Sonal Verma.
Unit 9: Vectors, Matrices and
TransformationsBy: Sushy Balraj and Sonal Verma
★ Any quantity that contains both length and direction.
★ It is named from the initial to the terminal point.
Vectors
2 right
3 up
Name:
➔ Example 1: ⇀
★ 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).
Component Form: <2,3>
OA
➢ Common Mistake: People forget to put the brackets around the component form of the vector.
Vectors
★ 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?
BC: <16-12, 2-4> = <4, -2>AB: <12-0, 4-0> = <12,4>
★ Rectangular array of elements.★ Arranged in rows and columns.
Matrices
(3x3), (3x3)
★ In order to add or subtract matrices, the dimensions need to be the same.
★ Example 2:
★ Multiplying matrices…
Matrices
➢ Common Mistake: People often forget the prerequisites to add, subtract and multiply the matrices.
Real Life Example of Matrices
STEP 1 STEP 2
Examples of Matrices
A+B=
(2x4) (4x3)
★ Sliding a figure from one position to another.★ The shape and size stays congruent to the
original figure.
Translations
★ Use motion rules, component forms, or vectors to indicate translation.
(x, y) (x-5, y-2)
★ Units move up (+), down (-), left (-), right (+).
Common Mistake: People forget to put the negative behind the coordinate may lead to an incorrect translation.
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)
★ 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]
Rotations
➢ Use matrix multiplication to figure out the points for rotation by multiplying the matrix by the coordinates.
Rotation
➢ Struggle: Memorizing the coordinate rules for rotations.
➢ Common Mistake: 90° CW is actually 270° CCW!
★ stretch or shrink ★ k= scale factor★ enlargement= k>1★ reduction= 0<k<1
Dilations
➔ Example 3: Use the given point and k=2 to dilate the figure. Remember to use a ruler to get precise 📏
measurements!
★ combining transformations★ translation + reflection = glide reflection
Composition of Transformations
➔ 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)
➔ 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 2Reflection: across the y-axis
★ Different ways to approach transformations
Buried Treasures and Clock Problems
➔ Example 6:Start: (6,-1)Translate: (x+2, y-1)Reflect: x-axisRotate: 180°Component: <-2, 1>Where is the treasure?
➔ Example 5:
Start: 10Rotate: 180°Reflect: x-axis Rotate: 150°CCWWhat time is it?
Now there’s no possible way you
could fail this unit :)