Math 312 Bard College

Homework 2

Due Date: Friday, September 22

Instructions: Feel free to work together with other students in the class, though you must turn in your own copy of the solutions, and you must acknowledge anyone that you worked with.

  1. The following picture shows a parabola in \(\mathbb{R}^2\) whose vertex is at the origin and whose axis is the line \(y=2x\).
    1. Find the matrix for the counterclockwise rotation of \(\mathbb{R}^2\) that takes the line \(y=2x\) to the \(y\)-axis.
    2. Use your answer to part (a) to find the slope of the tangent line to the parabola at the point \((3,1)\).
  2. Let P be the plane \(x+y+z=0\) in \(\mathbb{R}^3\).
    1. Express each of the standard basis vectors \(\mathbf{i},\mathbf{j},\mathbf{k}\) as the sum of a vector normal to P and a vector parallel to P.
    2. Use your answer to part (a) to find the matrix for the reflection of \(\mathbb{R}^3\) across the plane P.
    3. Now let P be any plane through the origin in \(\mathbb{R}^3\), and let \((a,b,c)\) be a unit normal vector for P. Use the method of parts (a) and (b) to find a general formula for the matrix corresponding to the reflection of \(\mathbb{R}^3\) across P.
  3. As we discussed in class, every rotation matrix \(R\) in three dimensions can be written as a product of the form \[ R \,=\, \begin{bmatrix}\cos\gamma & -\sin\gamma & 0 \\ \sin\gamma & \phantom{+}\cos\gamma & 0 \\ 0 & \phantom{+}0 & 1\end{bmatrix} \begin{bmatrix}1 & 0 & \phantom{+}0 \\ 0 & \cos\beta & -\sin\beta \\ 0 & \sin\beta & \phantom{+}\cos\beta \end{bmatrix} \begin{bmatrix}\cos\alpha & -\sin\alpha & 0 \\ \sin\alpha & \phantom{+}\cos\alpha & 0 \\ 0 & \phantom{+}0 & 1\end{bmatrix}. \] The angles \(\alpha\), \(\beta\), and \(\gamma\) are called Euler angles for the rotation.
    1. Assuming \(\beta\in [0,\pi]\), show that the angle between the vectors \(\mathbf{k}\) and \(R\hspace{0.083333em}\mathbf{k}\) is equal to \(\beta\).
    2. Let \(\mathbf{n} = \mathbf{k}\times (R\hspace{0.083333em}\mathbf{k})\). Assuming \(\mathbf{n} \ne \mathbf{0}\) and \(\alpha,\beta,\gamma\in[0,\pi]\), find the angle between the vectors \(\mathbf{n}\) and \(\mathbf{i}\), as well as the angle between the vectors \(\mathbf{n}\) and \(R\hspace{0.083333em}\mathbf{i}\).
    3. Use parts (a) and (b) to find the Euler angles for the rotation determined by the matrix \[ \frac{1}{2}\begin{bmatrix}-1 & -\sqrt{2} & \phantom{+}1 \\ -1 & -\sqrt{2} & \phantom{+}1 \\ \sqrt{2} & \phantom{+}0 & \sqrt{2}\end{bmatrix}. \]