Notes on Harvard "Notes on spherical geometry" - Printable Version +- OMath! (http://math.elinkage.net) +-- Forum: Math Forums (/forumdisplay.php?fid=4) +--- Forum: Geometry (/forumdisplay.php?fid=6) +--- Thread: Notes on Harvard "Notes on spherical geometry" (/showthread.php?tid=720) Notes on Harvard "Notes on spherical geometry" - elim - 11-07-2016 03:18 PM 1. Some tools Trig identities$\underset{\,}{\,}$ $(1)\quad\cos(a\pm b)=\cos(a)\cos(b)\mp \sin(a)\sin(b)$ $(2)\quad\cos(X)-\cos(Y)=-2\sin\frac{X+Y}{2}\sin\frac{X-Y}{2}$ Cross products$\underset{\,}{\,}$ For vectors $A,\,B\in\mathbb{R}^3,\;$the cross-product $A\times B\;$ can be characterized geometrically by the conditions: $\bullet\quad (A\times B = 0)\iff (A,\,B\text{ are linearly dependent})$ $\bullet\quad (A\times B)\cdot A = (A\times B)\cdot B = 0$ $\bullet\quad A,\,B,\,A\times B\,$form a right-handed basis; $\bullet\quad |A\times B| = |A||B|\sin \widehat{A,B}$ Algebraically, if $A=(a_j)_{1\times 3},\,B = (b_j)_{1\times 3},\underset{\,}{\,}$then $A\times B = {\small\begin{pmatrix}a_2 b_3-a_3 b_2\\ a_3b_1-a_1b_3\\a_1b_2-a_2b_1 \end{pmatrix}} = (a_{j+1}b_{j+2}-a_{j+2}b_{j+1})_{1\times 3}\underset{\,}{\,}$ Where the subscript addition is defined by $\qquad i+j =\min\{k\in\mathbf{N}^+:\; i+j \equiv k\pmod{3}\}$