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

  1. 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 \(\begin{pmatrix} 0 \\ 2 \\ 0 \end{pmatrix}\).

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