Resource for all this:
Starter: Reprise 2×2 determinants
Algorithm: 3×3 determinants
Invariance of determinants.
INVERSES AND TRANSPOSES OF 3×3 MATRICES
Starter: matrix×matrix of cofactors
Exposition: inverting matrices
Exercise on determinant rules
3×3 MATRICES AS LINEAR TRANSFORMATIONS IN 3D
Starter: Reprise 2×2 matrices as linear transformations in 2D
Exposition: Equations of planes and lines in 3D
Equations of lines: write down the typical point for each of the lines in Ex.5D Q.1
Note that there are lots of different valid answers
Equations of planes: write down the typical point for each of the planes in Ex.5E Q.3
Note that there are lots of different valid answers
Exposition: Transformations in 3D. Look at Examples 17 and 18.
Look at Example 16 p.157 and do Ex.6D Q.4. Look at Example 17 p.157 and do Ex.6D Q.7.
Look at Example 18 p.158 and show that the image is a line without doing more working than the bottom of p.158.
Ignore the nonsense on p.159 of the textbook, which includes dividing by zero.
Look at Example 21 on p.163. Do Ex.6E Q.7. Do Ex.6E Q.1
Activity: Ex.6D, 6E
LESSON 3: EIGENVALUES AND EIGENVECTORS
Ax = λx
det (A−λI) = 0
Activity: Ex.6F Q.1-8
Exposition: Eigenvalues in engineering
Discussion: what if complex eigenvalues?
In FP1 we found out what (some) matrices looked like by plotting where they mapped the points (1,0) and (0,1) to.
Here a computer has taken a matrix and shows where it maps a whole heap of other points on the unit circle (as well as the points (1,0) and (0,1): the red line is what the line-segment from (0,0) to (1,0) maps to, and the blue line is what the line-segment from (0,0) to (0,1) maps to).
and draw in new axes along the lines you can see in the picture (lines in the direction of the eigenvectors) where the matrix just maps every point on the line further out (in this case; or could be closer in, for an another matrix) along the same line.
The geometrical transformation is still the same, but with different axes the matrix (table of numbers) representing it will be different.
With the new axes, the matrix is just a scaling along the two axes (no shearing).
It is a diagonal matrix: every element in it is zero except the leading diagonal (and the elements there are the eigenvalues).
In fact, it is just this (if we swivel the picture so that the axes are straight across and straight up).
We have simplified both the description of the geometrical transformation (to just a scaling, possibly by different amounts, in two perpendicular directions), and the matrix for it (to a diagonal matrix).
For a matrix A, the new axes along the eigenvector directions can be described by a matrix P, where P is a matrix formed by putting the normalised eigenvectors for A, one by one, into the columns of P.
The change of axes changes the table of numbers (matrix) describing the same transformation to P−1AP
P−1AP = D, which is a diagonal matrix consisting of the eigenvalues of A down the leading diagonal, and zeroes everywhere else.
The normalised eigenvectors can be in any order in P, and the eigenvalues in any order in D, but the eigenvectors and eigenvalues have to be in the same order (first eigenvalue is the eigenvalue corresponding to first eigenvectors, second eigenvalue is the eigenvalue corresponding to the second eigenvector, and so on.
This calculation is especially useful and easy if A is a symmetric matrix. Then, amazingly, all the eigenvalues are real, and the eigenvectors are all perpendicular (orthogonal) to each other. PTP=I (if you’re asked to verify that eigenvectors are orthogonal, you do it by verifying PTP=I), and so P−1=PT. In other words the inverse is very easy to calculate.
PTAP = D
This special case of symmetric matrices is important in real life, and in FP3 you deal with diagonalisation only for symmetric matrices.
Homework: Ex.6G Q.2, 3, 6, 9, 10. Review Questions p.196 Q.24. Ex.6F Q.2(a), 3, 4
LESSON 4: DOT PRODUCT AND CROSS PRODUCT
Cross product in motor effect
Calculating cross product
Resource: Wolfram CDF: cross product of vectors
Homework: Ex.5A Q.1-6, Ex.5B Q.1-6
NOTE ON QUATERNIONS
Dot product and cross product come from the multiplication of four-dimensional numbers (quaternions). The product of two all-“imaginary” quaternions
(a1i+b1j+c1k)(a2i+b2j+c2k) = −a.b + a×b
This explains why dot product and cross product “work” fairly well despite lacking many features of “proper” multiplication operations, and why, despite being completely different, they have some subtle links.
Quaternions can also be represented as 2×2 matrices of complex numbers.
Resource: Videos on quaternions
AREAS, VOLUMES, AND TRIPLE PRODUCT
Cross product as area of parallelogram
Triple product as volume of parallelepiped. Volume of tetrahedron.
Classwork and homework: Ex.5B Q.6, 7, 8, 9, 12 (areas)
Ex.5C Q.1, 2, 3, 5, 6 (volumes)
LESSON 5-6: USES OF VECTORS IN 3D GEOMETRY
Starter: reprise equations of lines and planes in 3D
Ex.5D Q.1, 2, 3 (equations of lines)
Ex.5E Q.1, 3, 4 (equations of planes)
Resources: Notes on methods different from the book
Line of intersection of two planes: Ex.5F Q.3, Ex.5G Q.9 (a), (b)
Angle between planes: Ex.5F Q.4,5; Ex.5G Q.11. Angle between lines Ex.5G Q.16
Angle between line and plane: Ex.5F Q.8; Ex.5G Q.12
Distance from point to plane: Ex.5F Q.9, Ex.5G Q.6, Q.13 (distance between a line parallel to a plane, and the plane, is the same as the distance from any point – might as well be the “a” point – on the line to that plane); Review Exercises 2 Q.16 p.195
Distance between parallel planes: Ex.5F Q.10
Distance between skew lines: Ex.5F Q.11, Ex.5G Q.1, 2, 3
Distance from point to line, and distance between parallel lines: Ex.5F Q.12; Review Exercises 2 Q.19 p.196