# 19 Further Pure 4

Candidates will be expected to be familiar with the knowledge, skills and understanding implicit in the modules Core 1, Core 2, Core 3, Core 4 and Further Pure 1.

Candidates may use relevant formulae included in the formulae booklet without proof.

 Definition and properties of the vector product. Calculation of vector products. Including the use of vector products in the calculation of the area of a triangle or parallelogram. Calculation of scalar triple products. Including the use of the scalar triple product in the calculation of the volume of a parallelepiped and in identifying coplanar vectors. Proof of the distributive law and knowledge of particular formulae is not required. Applications of vectors to two- and three-dimensional geometry, involving points, lines and planes. Including the equation of a line in the form $({\bf r - a}) \times {\bf b} = 0$. Vector equation of a plane in the form ${\bf r.n} = d$ or $\bf{\mbox{r}} = \mbox{a}+ { \lambda } \bf{ \mbox{b} } + \mu \bf{ \mbox{c}}$ . Intersection of a line and a plane. Angle between a line and a plane and between two planes. Cartesian coordinate geometry of lines and planes. Direction ratios and direction cosines. To include finding the equation of the line of intersection of two non-parallel planes. Including the use of $l^2 + m^2 + n^2 = 1$ where $l, m, n$ are the direction cosines. Knowledge of formulae other than those in the formulae booklet will not be expected.
 Matrix algebra of up to 3 x 3 matrices, including the inverse of a 2 x 2 or 3 x 3 matrix. Including non-square matrices and use of the results $({\bf AB})^{-1} = {\bf B}^{-1}{\bf A}^{-1}$ and $({\bf AB}{^T}) = {\bf B}{^T}{\bf A}{^T}$ Singular and non-singular matrices. The identity matrix $\bf{I}$ for 2 x 2 and 3 x 3 matrices. Matrix transformations in two dimensions: shears. Candidates will be expected to recognise the matrix for a shear parallel to the $x$ or $y$ axis. Where the line of invariant points is not the $x$ or $y$ axis candidates will be informed that the matrix represents a shear. The combination of a shear with a matrix transformation from MFP1 is included. Rotations, reflections and enlargements in three dimensions, and combinations of these. Rotations about the coordinate axes only. Reflections in the planes $x = 0, y=0, z=0, x=y, x=z, y=z$ only. Invariant points and invariant lines. Eigenvalues and eigenvectors of 2 x 2 and 3 x 3 matrices. Characteristic equations. Real eigenvalues only. Repeated eigenvalues may be included. Diagonalisation of 2 x 2 and 3 x 3 matrices. $\bf{M} = \bf{UDU}^{-1}$ where $\bf{D}$ is diagonal matrix featuring the eigenvalues and $\bf{U}$ is a matrix whose columns are the eigenvectors. Use of the result $\bf{\mbox{M}}^n = \bf{\mbox{UD}}^n \bf{\mbox{U}}^{-1}$
 Consideration of up to three linear equations in up to three unknowns. Their geometrical interpretation and solution. Any method of solution is acceptable.
 Second order and third order determinants, and their manipulation. Including the use of the result ${det} ({\bf AB}) = {det} {\bf A} {det} {\bf B}$, but a general treatment of products is not required. Factorisation of determinants. Using row and/or column operations or other suitable methods. Calculation of area and volume scale factors for transformation representing enlargements in two and three dimensions.
 Linear independence and dependence vectors.