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TRANSFORMATION IN THE PLANE
Transformation in the plane is a mapping which shifts an object from one position to another within the same plane.
Examples of transformations in the xy plane are
i. Reflection
ii. Rotation
iii. Enlargement
iv. Translation

REFLECTION
-The action or process of sending back light, heat or sound from a surface.

ISOMETRIC MAPPING
-Is a transformation which the object size is maintained.
-Reflection is an example of isometric mapping.

Reflection in the line included an angle (α) passing through the origin.
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE

EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE inclined at B with the coordinates being (X, Y)
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE is the image of EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE under reflection is EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE
PP is perpendicular to OS (EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE is the line of reflection)
POS= α EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE β

Δ OPQ is right angled a
t Q

Hence X = OP cos B……(i)
Y = OP sin B
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE is perpendicular to the axis of X at R

Coordinates of R are ( X1,0)
OR= X 1
RP1=Y1
Δ OP1R is right angled at R
P1OS= POS= α-β
Angle P1OR = α – β + α – β + β = 2α – β
Cos (2α-β)= X1
OP1
X1= OP1cos (2α-β)….. (iii)

Y1 = sin (2α-β)
OP1
Y1=OP1sin (2α-β)……(iv)
Cos (A+B) = cos A cos B – sin A sin B
Sin(A+B) = sin A cos B+ sin B cos A
X1 = OP1 cos 2α cosβ + OP sin 2α cos β…. (iii)
Y1= OP sin 2α cos β – OP1 sin B cos 2α
X1= OP cos 2α cos β+ OP sin 2α sin β
Y1= OPsin2αcos β – Op sinβ cos 2α
X1 = OP cos β cos2α + OP sin β sin 2α
Y1= OP cos βsin 2α – OP sin β cos 2α
X1 = X cos 2α + Y sin 2α …..( i)
Y1 = X sin 2α – Y cos 2 α …..(ii)
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE= EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE


Exercise
1. Find the image of the point A (1, 2) after a reflection in the Y= X plane.
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE



Solution;
Y= X
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE = 1
Tan α =EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE = 1
α= 900
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE= EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE= EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE= EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE


(X1, Y1) = (2, 1)

2. Find the image of B (3,4) after a reflection in the line Y= -X followed by another reflection in the line Y= 0

My = -X
My = 0

Y= -X

EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE


y = x
y = 1
x
α = – 450 for clockwise movement
Or
α = 1350 anticlockwise movement

EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE = EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE

EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE = EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE

EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE = EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE

(X1, Y1) = (-4, -3)

Followed reflection at Y = 0
Tan α = 0
α = 0
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE = EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE

EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE= EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE

EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE= EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE

(X11, Y11) = (-4, 3)


In questions 3 to 6, write the matrix of reflection in a given line.
3. Y= 0 ( the X axis)
Y= 0
Tan α= 0
α= 0
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE= EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE

Mx = EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE


4. Y = X
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE = 1

Tan α= 1
Α= 900

EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE = EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE

= EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE


5. X = 0
Tan α = 0
α= 0
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE = EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE
= EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE

6. Find the image of the point (1, 2) after a reflection in the line Y= X followed by another reflection in the line Y= -X.
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE = 1
Tan α = 90 0
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE= EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE
= EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE
= EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE

= ( 2, 1)


ROTATION
Find the image of the point B (1,2) after a rotation by 900 about the origin.

Solution:

EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE = EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE = EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE
Rθ = EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE = EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE


IMAGE OF A POINT ROTATED AT INCLINED LINE AT ANGLE B
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE
Let EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE be inclined at an angle B
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE = EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE
PA is perpendicular to the X – axis at A
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE = X , EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE= Y
ΔOAP is right angled at A with POA = B
Cos B = X
OP
X = OP Cos B …….(i)
Sin B = Y
OP
Y = OP sin B ….. (ii)

EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE is perpendicular to the X – axis at B
Cos (B+ θ) = X1
OP
X 1 = EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE cos ( B + θ)
X1 = EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE cos B cos θ – OP sin B sinθ
X1 = X cos θ – Y sin θ ……(iii)
Sin (B+ θ) = Y 1
OP
Y1= EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE sin (B+ θ)
Y1= EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANEsin B cos θ + EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE sin θ cos B
Y1= Y cos θ + X sin θ …..(iv)

In matrix form
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE = EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE

Question
The matrix of rotation
1. 900 about the origin.
R = EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE

= EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE


2. find the image of (1,2) after a rotation of 900 followed by another rotation of 2700 about the origin.

Solution
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE = EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE

EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE = EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE = EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE

= EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE

3. Find the image of 3X + 4Y + 6 = 0 under a rotation of 900 about the origin.

Solution:
3X + 4Y = -6
X intercept, Y = 0
3X = -6
X = -2
Y intercept, X = 0
4Y= -6
Y = EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE
(-2,EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE)

EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE = EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE

= EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE

EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE = EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE



TRANSLATION
Exercise
1. A translation takes every point a distance 1 unit to the left and 2 units downwards. Find where it takes.
a. (0,0)
b. (1,1)
c. (3,7)

Solution
a. (0,0)

(a,b) = (-1,-2)
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE


b. (1,1)
(a,b) = (-1,-2)
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE


C. (3,7)
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE


2. If translation takes the origin to (8,7). Given

U= (-12, 12) , v= (6,-16)

find T(u+v)
= ( u + v)
EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE


3. Find the image if the line 3x + 4Y + 6 = 0 under a translation by the vector (-6,-1)
3x + 4Y -6 = 0

Solution

Y = mx + c

4Y = -3/4 x – 3/2
X intercept, y=0
3x+ 0 + 6=0
3X= -6
X = -2
(-2,0)

Y intercept, X= 0
0 + 4Y + 6 = 0
4Y = -6
Y = EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE
(0,EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE)

EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE

(-8,-1) and (-6,EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE)


Y- Y1 = M (X-X1)
Y- -1 = M( X- -8)
Y + 1 = EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE (X+8)
Y+1 = EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE X + EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE
Y= EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANEX EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE 7

4. Find the image of the line
Y = X under a translation by the vector (5,4)

Solution

Y = X
X
-2
1
Y
-2
1


EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE


(3,2) and (6,5)
Slope = EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE

= EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE

= EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE

M =1 (3 ,2)
Y-Y1 = M ( X –X1)
Y-2 = 1(X – 3)
Y-2 = X – 3
Y = X – 3 + 2
Y = X -1

LINEAR TRANSFORMATIONS
Consider transformation T,
Let u and v be two vectors
Let t be the real number
The t is a linear transformation if it obeys the following properties.
i. T (tu) = t T (u)
ii. T( u+v) = T (u) + T(v)


ENLARGEMENT
The transformation which magnifies an object such that its image is proportionally increased or decreased in size by some factor.
General matrix of enlargement is EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE

where k is non zero or real number (Linear scale factor)

EXERCISE
1. Find the image of (1,2) under the enlargement by T= EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE

Solution

EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE = EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE 1 = EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE

(X1, Y 1) = (5, 10)

2. Find the image of ( -1/2 , -1/3 ) under the enlargement by
T=EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE

Solution

EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE

(X, Y ) = (6, 4)

3. Find the enlargement matrix which maps the point (3,-4) into (18, -24).

EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE

-24 = 4k

18 = -3k

K= -6

-24 = 4K

K = -6

(-6,-6)




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1 Comment

  • EcoleBooks | MATHEMATICS O LEVEL(FORM FOUR) NOTES - TRANSFORMATION IN THE PLANE

    ajuna julius, March 22, 2024 @ 5:49 pm Reply

    thanks

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