From mboxrd@z Thu Jan 1 00:00:00 1970 Path: news.gmane.org!not-for-mail From: "Pascal J. Bourguignon" Newsgroups: gmane.emacs.help Subject: Re: why are there [v e c t o r s] in Lisp? Date: Sat, 17 Oct 2015 04:22:58 +0200 Organization: Informatimago Message-ID: <87lhb2cjy5.fsf@kuiper.lan.informatimago.com> References: <87oafzpqfj.fsf@debian.uxu> NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=utf-8 Content-Transfer-Encoding: 8bit X-Trace: ger.gmane.org 1445048724 16985 80.91.229.3 (17 Oct 2015 02:25:24 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sat, 17 Oct 2015 02:25:24 +0000 (UTC) To: help-gnu-emacs@gnu.org Original-X-From: help-gnu-emacs-bounces+geh-help-gnu-emacs=m.gmane.org@gnu.org Sat Oct 17 04:25:23 2015 Return-path: Envelope-to: geh-help-gnu-emacs@m.gmane.org Original-Received: from lists.gnu.org ([208.118.235.17]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1ZnHBS-0006xK-89 for geh-help-gnu-emacs@m.gmane.org; Sat, 17 Oct 2015 04:25:22 +0200 Original-Received: from localhost ([::1]:56672 helo=lists.gnu.org) by lists.gnu.org with esmtp (Exim 4.71) (envelope-from ) id 1ZnHBR-0001a0-7r for geh-help-gnu-emacs@m.gmane.org; Fri, 16 Oct 2015 22:25:21 -0400 Original-Path: usenet.stanford.edu!fu-berlin.de!uni-berlin.de!individual.net!not-for-mail Original-Newsgroups: gnu.emacs.help Original-Lines: 29 Original-X-Trace: individual.net uDdef8mPCuFEyl0Zcg+WiwlXPx+l4RUlOBzSaCXbrIPSfxP8cJ Cancel-Lock: sha1:MGM4ZjE2NzA4YTNmMGZhNDlmOTE0NTllYWI4ZDlmODdjODM1Y2RkYg== sha1:OBhJA6eeGBrDzf8vWeKjTA9z21M= Face: iVBORw0KGgoAAAANSUhEUgAAADAAAAAwAQMAAABtzGvEAAAABlBMVEUAAAD///+l2Z/dAAAA oElEQVR4nK3OsRHCMAwF0O8YQufUNIQRGIAja9CxSA55AxZgFO4coMgYrEDDQZWPIlNAjwq9 033pbOBPtbXuB6PKNBn5gZkhGa86Z4x2wE67O+06WxGD/HCOGR0deY3f9Ijwwt7rNGNf6Oac l/GuZTF1wFGKiYYHKSFAkjIo1b6sCYS1sVmFhhhahKQssRjRT90ITWUk6vvK3RsPGs+M1RuR mV+hO/VvFAAAAABJRU5ErkJggg== X-Accept-Language: fr, es, en User-Agent: Gnus/5.13 (Gnus v5.13) Emacs/24.3 (gnu/linux) Original-Xref: usenet.stanford.edu gnu.emacs.help:215404 X-BeenThere: help-gnu-emacs@gnu.org X-Mailman-Version: 2.1.14 Precedence: list List-Id: Users list for the GNU Emacs text editor List-Unsubscribe: , List-Archive: List-Post: List-Help: List-Subscribe: , Errors-To: help-gnu-emacs-bounces+geh-help-gnu-emacs=m.gmane.org@gnu.org Original-Sender: help-gnu-emacs-bounces+geh-help-gnu-emacs=m.gmane.org@gnu.org Xref: news.gmane.org gmane.emacs.help:107688 Archived-At: Aurélien Aptel writes: > I'm just nitpicking, but: > […] > In pure linear algebra, "direction" and "position" are not defined for > vectors. Well once you define dot-product, you have defined implicitely angles and therefore direction. A⋅B = |A|×|B|×cos(∠AB) A⋅B/(|A|×|B|) = cos(∠AB) For a vector V and a base (In), (cos(∠VIn))=(V⋅In/|V|) defines the direction of the vector V. If (In)=((δin)), then (V⋅In/|V|) = V/|V| which shows that any vector defines a direction by itself. The essence of vector is to define a direction and a magnitude. Notably for vector spaces without a finite basis, V/|V| is still the direction of the vector V, intrinsically (every unitary vector is a distinct direction). -- __Pascal Bourguignon__ http://www.informatimago.com/ “The factory of the future will have only two employees, a man and a dog. The man will be there to feed the dog. The dog will be there to keep the man from touching the equipment.” -- Carl Bass CEO Autodesk