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Notes on Harvard "Notes on spherical geometry" - Printable Version

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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
$\qquad i+j =\min\{k\in\mathbf{N}^+:\; i+j \equiv k\pmod{3}\}$