Homework 6, Due by email to TA Oct 30

This has lots of little math questions based on the lecture. They mostly require thinking, not just plugging in.

1. (1 pt) Write the 4x4 homogeneous matrix for a projection onto the viewplane {$x+2y+3z=6$} with the center of projection at (0,0,0).
2. (1 pt) Write the 4x4 homogeneous matrix for a projection onto the viewplane {$z=0$} with the center of projection at (-1,0,0).
3. (1 pt) Write the 4x4 homogeneous matrix for a uniform scale by 3 followed by a 3D translation by (1,1,1).
4. (2 pt) Write the 4x4 homogeneous matrix for a 3D translation by (1,1,1) followed by a uniform scale by 3.
5. (1 pt) Normalize the 3D Cartesian vector (1,2,3).
6. (2 pt) Write the vector equation for rotating points by {$180^\circ$} about the axis (3,0,4).
7. (1 pt) What does the point (0,2,0) rotate to?
8. (1 pt) If {$a=(4,5,6)$} then write {$a\cdot p a$} as a matrix depending on {$a$} times {$p$}.
9. (1 pt) Write {$a\times p$} ditto.
10. This {$M=\begin{pmatrix}.6&0&.8\\0&\color{red}{0}&0\\.8&0&-.6\end{pmatrix} $} is not a rotation matrix.
    1. (1 pt) Give 2 rules for being a rotation matrix that it violates.
    2. (1 pt) Change {$m_{2,2}$}, highlighted in red above, to make {$M$} a rotation matrix.
11. (2 pt) This {$M=\begin{pmatrix}-1&0&0\\0&1&0\\0&0&-1\end{pmatrix} $} is a rotation matrix. What are the normalized axis and angle of rotation?