91Ó°ÊÓ

Let \(S\) be the sphere with radius 1 centered at \((0,0,1),\) and let \(S^{*}\) be \(S\) without the "north pole" point (0,0,2) . Let \((a, b, c)\) be an arbitrary point on \(S^{*}\). Then the line passing through (0,0,2) and \((a, b, c)\) intersects the \(x y\) -plane at some point \((x, y, 0)\), as in Figure 1.6.10. Find this point \((x, y, 0)\) in terms of \(a, b\) and \(c\). (Note: Every point in the \(x y\) -plane can be matched with a point on \(S^{*}\), and vice versa, in this manner. This method is called stereographic projection, which essentially identifies all of \(\mathbb{R}^{2}\) with a "punetured" sphere.)

Write the normal form of the plane containing the given points. $$ (1,0,3),(1,2,-1),(6,1,6) $$

Prove the Jacobi identity: \(\mathbf{u} \times(\mathbf{v} \times \mathbf{w})+\mathbf{v} \times(\mathbf{w} \times \mathbf{u})+\mathbf{w} \times(\mathbf{u} \times \mathbf{v})=\mathbf{0}\)