Geometric Algebra HW 1 (Wedge Product)
MultiV 2021-22 / Dr. Kessner

1. For each of the following pairs of vectors \(\mathbf{u}\) and \(\mathbf{v}\), find the wedge product \(\mathbf{u} \wedge \mathbf{v}\). Draw the vectors and make sure your answer makes sense geometrically.

   1. \(\mathbf{u} = \begin{pmatrix} 2 \\ 0 \end{pmatrix}\), \(\mathbf{v} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}\)

   2. \(\mathbf{u} = \begin{pmatrix} 0 \\ 2 \end{pmatrix}\), \(\mathbf{v} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}\)

   3. \(\mathbf{u} = \begin{pmatrix} 3 \\ 0 \end{pmatrix}\), \(\mathbf{v} = \begin{pmatrix} 3 \\ 3 \end{pmatrix}\)

   4. \(\mathbf{u} = \begin{pmatrix} 3 \\ 1 \end{pmatrix}\), \(\mathbf{v} = \begin{pmatrix} 1 \\ 3 \end{pmatrix}\)

2. Find the area of the triangle determined by the two vectors \(\mathbf{u} = \begin{pmatrix} 2 \\ 2 \end{pmatrix}\) and \(\mathbf{v} = \begin{pmatrix} -2 \\ 2 \end{pmatrix}\).
   Find a general formula for the area of a triangle determined by two vectors \(\mathbf{u}\) and \(\mathbf{v}\).

3. Find the distance from the point \((2,2)\) to the line \(2x+2y=2\).

4. Find the distance from the point \((7,7)\) to the line \(6x + 8y = 48\).

Answers:
1a. \(2 \mathbf{e_1}\wedge\mathbf{e_2}\)
1b. \(-2 \mathbf{e_1}\wedge\mathbf{e_2}\)
1c. \(9 \mathbf{e_1}\wedge\mathbf{e_2}\)
1d. \(8 \mathbf{e_1}\wedge\mathbf{e_2}\)
2. \(A = \frac{1}{2}|\mathbf{u}\wedge\mathbf{v}| = \frac{1}{2}(8) = 4\)
3. \(\frac{3\sqrt{2}}{2}\)
4. \(5\)