Matrices and Transformation
- A vector quantity has both magnitude and direction. For example:
- Vectors a and b represented by the line segments can be added using the parallelogram rule or the nose-to-tail method
- Multiplication by a scalar:
- A scalar quantity has a magnitude but no direction.
- The negative sign reverses the direction of the vector.
- Column vector:
- The top number is the horizontal component and the bottom component is the vertical component.
- Parallel vectors:
- Vectors are parallel if they have the same direction.
- In general, the vector is parallel to
- Modulus of a vector:
- In general, if
- Multiplication by scalar:
- Multiplication by vector:
- You can only multiply if the number of columns on the left matrix is equal to the number of rows on the right matrix.
- Determinant = leading diagonal – secondary diagonal
- To work out the inverse of a matrix, alternate the leading diagonal’s elements, negate the secondary diagonal, and multiply by 1/determinant
- Reflection (M):
- When describing a reflection, the position of the mirror line is essential.
- Rotation (R):
- To describe a rotation, the center of rotation, the angle of rotation, and the direction of the rotation are required.
- A clockwise rotation is negative and an anticlockwise rotation is positive.
- Translation (T):
- When describing a translation, it is necessary to give the translation vector
- Enlargement (E):
- To describe an enlargement, state the scale factor,k, and the center of enlargement.
- If k > 0, both object and image lie on the same side of the center of enlargement.
- If k < 0, object and image lie on opposite sides of the center of enlargement.
TRANSFORMATION BY MATRICES
- — Reflection in the x – axis
- — Reflection in the y – axis
- — Reflection in the line y = -x
- — where k = scale factor and center of enlargement = (0, 0)
- — Rotation 90° anticlockwise, center (0, 0)
- — Rotation 90° clockwise, center (0, 0)
- — Rotation 180° clockwise/anticlockwise, center (0, 0)