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#### Matrices and Transformation

##### VECTORS
• 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
##### MATRICES
• 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:
• Determinant = leading diagonal – secondary diagonal
• Inverse:
• To work out the inverse of a matrix, alternate the leading diagonal’s elements, negate the secondary diagonal, and multiply by 1/determinant
##### TRANSFORMATION
• 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:
• — Reflection in the x – axis
• — Reflection in the y – axis
• — Reflection in the line y = -x
• Enlargement:
• — where k = scale factor and center of enlargement = (0, 0)
• Rotation:
• — Rotation 90° anticlockwise, center (0, 0)
• — Rotation 90° clockwise, center (0, 0)
• — Rotation 180° clockwise/anticlockwise, center (0, 0)