Applications of Matrices in Coordinate Geometry and Transformations
1. Finding the Area of a Triangle
Given three vertices , the area of the triangle can be found using determinants:
where you choose the sign so that the area is positive.
This determinant-based approach is useful in computational geometry and eliminates the need for direct distance calculations.
2. Testing for Collinear Points
Three points are collinear if the determinant of the matrix is zero:
3. Two-Point Form of the Equation of a Line
The equation of a line passing through two points and can be written using determinants:
4. Area of a Quadrilateral
Given four vertices , , , and , the area of the quadrilateral can be found by splitting it into two triangles and applying the determinant formula separately to each.
where you choose the sign so that the area is positive.
5. Area of a Parallelogram
The area of a parallelogram with vertices , , , and is:
is the absolute value of the determinant.
where .
6. Transformation Matrices
Transformation matrices help modify shapes and points in the coordinate plane.
Reflection
Across the x-axis:
Across the y-axis:
Across the line :
Across the line :
Rotation Around the Origin
Counterclockwise:
Counterclockwise:
Counterclockwise:
Scaling by Factors
Horizontal scaling by factor :
Vertical scaling by factor :
7. Cryptography Using Matrices
Matrices are used in cryptography for encoding and decoding messages using key matrices. If is the plaintext matrix and is the key matrix, encryption is done using matrix multiplication: