The Translation is a transformation in which each point of the given object is displaced through definite distance and direction. The displacement is defined by a translation vector (a, b).

The following are the properties of translation:

1.   The object and the image under the translation are congruent.

2.   The lines joining any point of the object with its corresponding image are parallel and equal.

Rotation is a transformation in which each point on the object is rotated through an angle about a fixed point. The fixed point is called the centre of rotation and the angle is called the angle of rotation.

There are two types of rotations on the basis of directions:

1. Anticlockwise or Positive (+) rotation

2. Clockwise or Negative (–) rotation

Reflection is a transformation which has a mirror image about a line as a two way mirror. A line which plays the role of two way mirror to give the image of given object is called the axis of reflection.

The following are the properties of reflection:

a.   The object and the image under the reflection are congruent.

b.   The object and the image will be at equal distances from the axis of reflection.

c.   The line joining the object and its image will be perpendicular to the axis of reflection.

Transformation in sense of mathematics is the change of position, shape, or size of a geometric figure. A figure before the transformation is called an object and after transformation is called the image of the original object.

In transformation, every point A of an object maps with a unique and distinct point A’ of the image, so it is a one-to-one function. 

If T is a transformation and A’ is the image point of A under T, then we denote it by T(A) = A’ or by T: A → A’ or simply by A → A’. 

If another transformation brings the image A’ back to the object (pre-image) point A, then such a transformation is called the inverse of the transformation T and it is denoted by T^-1.

Transformation sometimes leaves certain points unchanged. These points are said to be invariant under the given transformation. Any other features which are left unaltered by the transformation such as distance, angle, parallelism, etc. are also described as invariant.

Types of Transformation

If a transformation maps a point A to A itself, then it is called an Identity Transformation and it is denoted by I. So, I(A) = A. Identity transformation leaves all points unchanged.

If a transformation does not change the figure after transformation, then it called an Isometric Transformation. Isometric transformation maps a figure to a congruent image. It means object figure and image figure are congruent under the isometric transformation.

If a transformation changes the figure after transformation, then it is called a Non-Isometric Transformation. Non-isometric transformation maps a figure to a similar image. It means object figure and image figure are similar under non-isometric transformation.

There are four fundamental transformations:

A.   Reflection

B.   Rotation

C.   Translation

D.  Enlargement (or Reduction)

Reflection, Rotation, and Translation are isometric transformations and Enlargement (or Reduction) is a non-isometric transformation.

Study the following figures for the general concepts of Reflection, Rotation,  Translation, Enlargement, and Reduction. 

Reflection

Rotation

Enlargement

Reduction

Visit the following pages for specific study on particular topics of transformation and related worked out examples:

- Reflection

- Rotation

- Translation

- Enlargement (or Reduction)

Do You Have Any Question?

 

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Vector Geometry: We can study different properties and relations relating to geometry with the help of vectors. Such a study is known as the Vector Geometry.

Here are 12 geometrical theorems and their proofs by vector method.

Product of Vectors: Product of vectors or multiplication of vectors can be performed by the following two ways:

• Scalar Product or Dot Product
• Vector Product or Cross Product

Vector Operations: The vectors can be added, subtracted or multiplied with one another. Addition of vectors, difference of vectors, multiplication of a vector by a scalar, product of vectors, etc. are called vector operations.

 

Types of Vectors: After we have learnt ‘Vectors and its Components’ and ‘Magnitude and Direction of a Vector’, we’ll learn the Types of Vectors here. Vectors are of different types according to their magnitude, direction, position with respect to other given vector, and by the method of representing vectors.

Following are the different types of vectors:

Magnitude of a Vector 

Magnitude of a vector is the length of its directed line segment from the initial point to the terminal point. So, the magnitude of a vector is a positive number which is the measure of the line segment representing that vector. Magnitude of a vector is also known as the modulus or the absolute value of the vector.