Geometric Algebra HW 3 (Wedge Product in \(\mathbb{R}^3\))
MultiV 2021-22 / Dr. Kessner

1. For each of the following sets of vectors, find the following: \(u \wedge v\), \(u \times v\), \(u \wedge v \wedge w\), and \((u\times v)\cdot w\).

   1. \(u = 3e_1\), \(v = 2e_2\), \(w = 5e_3\)

   2. \(u = 3e_1 + e_2\), \(v = 2e_1 + 2e_2\), \(w = 5e_3\)

   3. \(u = 3e_1 + e_2\), \(v = 2e_1 + 2e_2\), \(w = 7e_1 + 11e_2 + 5e_3\)

2. Use the wedge product representation of the plane \[(r-r_0)\wedge u \wedge v = 0\] to solve the following problems.

   1. Find the standard equation of the plane through the points \(\begin{pmatrix} 10 \\ 0 \\ 0 \end{pmatrix}\), \(\begin{pmatrix} 0 \\ 10 \\ 0 \end{pmatrix}\), and \(\begin{pmatrix} 0 \\ 0 \\ 10 \end{pmatrix}\).
      Also find the distance from the plane to the origin.

   2. Find the standard equation of the plane through the points \(\begin{pmatrix} 4 \\ 0 \\ 0 \end{pmatrix}\), \(\begin{pmatrix} 4 \\ 4 \\ 0 \end{pmatrix}\), and \(\begin{pmatrix} 0 \\ 0 \\ 4 \end{pmatrix}\).
      Also find the distance from the plane to \(\begin{pmatrix} 0 \\ 2 \\ 0 \end{pmatrix}\).

Answers:

• 1a. \(u \wedge v = 6 e_1 e_2\), \(u \times v = 6e_3\), \(u \wedge v \wedge w = 30 \, e_1 e_2 e_3\), \((u\times v)\cdot w = 30\).

• 1b. \(u \wedge v = 4 e_1 e_2\), \(u \times v = 4 e_3\), \(u \wedge v \wedge w = 20 \, e_1 e_2 e_3\), \((u\times v)\cdot w = 20\).

• 1c. \(u \wedge v = 4 e_1 e_2\), \(u \times v = 4 e_3\), \(u \wedge v \wedge w = 20 \, e_1 e_2 e_3\), \((u\times v)\cdot w = 20\).

• 2a. \(x + y + z = 10\), \(d = \frac{10}{\sqrt{3}}\)

• 2b. \(x + z = 4\), \(d = 2\sqrt{2}\)