Math about vector and transformatio of plane it give a chance to understand

1. Vectors and Transformation of
the Plane
By zerubabel

2. Vector
• A vector is a quantity that has both magnitude
(length) and direction.
• Vectors are often represented by directed line
segments or as ordered pairs in coordinate form.​
• Notation: Vectors are denoted by bold letters (e.g.,
v) or with an arrow over a letter (e.g., ).​
v⃗
• Zero Vector: A vector with zero magnitude and no
specific direction, denoted as .​
0⃗
• Unit Vector: A vector with a magnitude of 1, often
used to indicate direction.​

3. Vector Representation in the Plane
• Vectors in a 2D plane can be represented as
ordered pairs:​
• If point A has coordinates ( x1 , y1 ) and point B
has coordinates ( x2, y2 ) , then the vector from
A to B is:​A =(x2−x1,y2−y1)
B⃗
• Example:
• Let A = (2, 3) and B = (5, 7). Then:​
• A =(5−2,7−3)=(3,4)
B⃗

4. Vector Operations
a) Addition and Subtraction
Addition: Add corresponding components.​
+ =(u1+v1,u2+v2)​
u⃗ v⃗
Subtraction: Subtract corresponding components.​
− =(u1−v1,u2−v2)
u⃗ v⃗
Example:
Let =(4,−2) and =(1,3) , Then:​
u⃗ v⃗
+ =(4+1,−2+3)=(5,1)
u⃗ v⃗
− =(4−1,−2−3)=(3,−5)
u⃗ v⃗

5. b) Scalar Multiplication
Multiplying a vector by a scalar k scales its
magnitude by k and may reverse its direction if
∣ ∣
k is negative.​
k =(kv1,kv2) ​
v⃗
Example:
Let =(2,−3) and k=−2 . Then:​
v⃗
k =(−2)(2,−3)=(−4,6)
v⃗

6. Magnitude and Direction

7. Transformation of the Plane
• A transformation is a rule or function that
takes a point (or shape) in the coordinate plane
and maps it to another point.
• It changes the position, size, or orientation of
the figure.
• There are 3 main types of transformations:
1. Translation
2. Reflection
3. Rotation

8. 1. Translation
• A translation moves every point of a shape the
same distance in the same direction.
• A point ( x , y) becomes ( x + a , y + b ) , where a
and b are the horizontal and vertical shifts.
• Example:
Translate the point (2,3) by a=4a , b=−2b :
( 2 + 4 , 3 − 2 ) = ( 6 , 1 )
→

9. Translation
a) Dot: A single point (x, y) is moved to a new
location by adding a fixed value:
→ (x + a, y + b)
Example: (2, 3) translated by (4, -1) (6, 2)
→
b) Line: Every point on the line moves the same
way, so the line keeps its shape and direction,
just shifts position.
Result: Line remains parallel to original.
c) Plane: The entire plane shifts in one direction,
with no change in size or shape of any figures.

10. 2. Reflection (Flipping)
• A reflection flips a shape over a line (the mirror
line), such as the x-axis or y-axis.
• Common reflections:
• Over x-axis: (x,y) (x,−y)
→
• Over y-axis: (x,y) (−x,y)
→
• Over the line y=x : (x,y) (y,x)
→

11. • If the line is not on the axis of reflection, it gets
flipped over that axis.
• If the line lies on the mirror line, it maps to
itself.
• Result: Slope may change (except when reflected
over axes).

12. 4. Rotation (Turning)
• A rotation turns a point around a fixed center
(usually the origin) by a specific angle.
• Standard rotations around the origin:
• 90° counterclockwise: (x,y) (−y,x)
→
• 180°: (x,y) (−x,−y)(
→
• 270° counterclockwise (or 90° clockwise):
(x,y) (y,−x)
→

13. • Example:
Rotate (2,5) 90° counterclockwise:
(−5,2)
→
• Example 1:
Line y=x rotated 90° becomes y=−x
→
• Example 2:
Line through (0,0) and (1,2) rotated 180° line
→
through (0,0) and (-1, -2)