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乘法思维图：多边形与向量的几何演示

乘法思维图：多边形与向量的几何演示
几何画板及绘图内容加载中，请稍等……
2020-08-28 05:55
多边形向量积分概率统计几何作图
本示例通过几何方式展示了乘法思维，利用向量和多边形的平移与连接，直观地呈现了乘法运算的结构关系。核心数学概念包括乘法原理、百分比变化、随机过程中的乘法模型，以及定积分的几何解释。作图中使用了 GeoGebra 的 Segment、Vector、Translate、PolyLine、Point 等命令，构建了从 f 到 A40 的多个点和向量路径。适合高中及以上学生使用，尤其适用于数学建模、概率统计和几何可视化课程。

GeoGebra 命令及参数

  1. Segment((0, 0, 0), (12, 0, 0)) → f
  2. Element(L_1, d) → text1
  3. Element(L_2, d) → text2
  4. Element(L_7, d) → text3
  5. Element(L_8, d) → text4
  6. Element(L_9, d) → text5
  7. Element(L_{10}, d) → text6
  8. Element(L_{11}, d) → text8
  9. Name(C8) → K1
  10. Element(L_{23}, h + 1) → text9
  11. Element(L_{28}, d) → ACM
  12. Vector(NA003, NA026) → u_1
  13. Vector(NA003, NA018) → v_1
  14. Vector(NA003, NA004) → w_1
  15. Vector(NA003, NA012) → h_1
  16. Vector(NA004, NA016) → j_1
  17. Vector(NA004, NA012) → k_1
  18. Vector(NA004, NA032) → l_1
  19. Vector(NA004, NA033) → m_1
  20. Vector(NA012, NA018) → i_1
  21. Vector(NA012, NA013) → n_1
  22. Vector(NA012, NA014) → p_1
  23. Vector(NA013, NA057) → q_1
  24. Vector(NA013, NA034) → s_1
  25. Vector(NA013, NA077) → t_1
  26. Vector(NA013, NA026) → u_2
  27. Vector(NA013, NA014) → v_2
  28. Vector(NA014, NA028) → w_2
  29. Vector(NA015, NA028) → a_2
  30. Vector(NA016, NA058) → b_2
  31. Vector(NA016, NA033) → c_2
  32. Vector(NA018, NA031) → d_2
  33. Vector(NA018, NA026) → e_2
  34. Vector(NA018, NA058) → f_2
  35. Vector(NA026, NA031) → g_2
  36. Vector(NA026, NA074) → h_2
  37. Vector(NA026, NA035) → i_2
  38. Vector(NA026, NA056) → j_2
  39. Vector(NA026, NA032) → k_2
  40. Vector(NA028, NA031) → l_2
  41. Vector(NA028, NA034) → m_2
  42. Vector(NA028, NA053) → n_2
  43. Vector(NA028, NA032) → p_2
  44. Vector(NA031, NA035) → q_2
  45. Vector(NA031, NA032) → r_2
  46. Vector(NA032, NA035) → s_2
  47. Vector(NA033, NA058) → t_2
  48. Vector(NA053, NA079) → u_3
  49. Vector(NA053, NA073) → v_3
  50. Vector(NA056, NA057) → w_3
  51. Vector(NA056, NA076) → a_3
  52. Vector(NA056, NA074) → b_3
  53. Vector(NA056, MG062) → c_3
  54. Vector(NA057, NA076) → e_3
  55. Vector(NA057, NA073) → f_3
  56. Vector(NA058, NA077) → d_3
  57. Vector(NA058, NA132) → g_3
  58. Vector(NA060, NA077) → h_3
  59. Vector(NA060, NA081) → i_3
  60. Vector(MG061, MG090) → j_3
  61. Vector(MG061, NA077) → k_3
  62. Vector(MG061, NA099) → l_3
  63. Vector(MG061, MG135) → m_3
  64. Vector(MG061, MG195) → n_3
  65. Vector(MG061, MG109) → p_3
  66. Vector(MG062, MG085) → q_3
  67. Vector(NA071, NA122) → r_3
  68. Vector(NA073, NA182) → s_3
  69. Vector(NA073, NA077) → t_3
  70. Vector(MG061, NA073) → u_4
  71. Vector(NA074, NA075) → v_4
  72. Vector(NA075, NA098) → w_4
  73. Vector(NA075, MG090) → a_4
  74. Vector(NA075, NA077) → b_4
  75. Vector(NA075, NA100) → c_4
  76. Vector(NA075, NA099) → d_4
  77. Vector(NA075, NA076) → e_4
  78. Vector(NA075, MG085) → f_4
  79. Vector(NA075, NA101) → g_4
  80. Vector(NA076, NA098) → h_4
  81. Vector(NA076, NA100) → i_4
  82. Vector(NA076, NA082) → j_4
  83. Vector(NA077, NA079) → k_4
  84. Vector(NA077, NA102) → l_4
  85. Vector(NA077, NA078) → m_4
  86. Vector(NA079, NA130) → n_4
  87. Vector(NA079, MG135) → p_4
  88. Vector(NA079, MG195) → q_4
  89. Vector(NA081, NA098) → r_4
  90. Vector(NA082, MG085) → s_4
  91. Vector(MG085, MG110) → t_4
  92. Vector(MG090, MG221) → u_5
  93. Vector(MG090, MG135) → v_5
  94. Vector(MG090, MG115) → w_5
  95. Vector(NA098, NA123) → a_5
  96. Vector(NA100, NA291) → b_5
  97. Vector(NA100, NA121) → c_5
  98. Vector(NA100, NA106) → d_5
  99. Vector(NA100, NA101) → e_5
  100. Vector(NA101, NA291) → f_5
  101. Vector(NA101, NA121) → g_5
  102. Vector(NA101, NA106) → h_5
  103. Vector(NA101, NA106) → i_5
  104. Vector(NA291, NA123) → j_5
  105. Vector(NA291, NA106) → k_5
  106. Vector(NA079, NA102) → l_5
  107. Vector(NA102, NA125) → m_5
  108. Vector(NA121, MG110) → n_5
  109. Vector(NA121, NA134) → p_5
  110. Vector(NA121, MG109) → q_5
  111. Vector(MG109, NA123) → r_5
  112. Vector(MG109, MG137) → s_5
  113. Vector(MG109, MG159) → t_5
  114. Vector(MG109, MG217) → u_6
  115. Vector(MG109, MG218) → v_6
  116. Vector(MG115, MG220) → w_6
  117. Vector(MG109, NA123) → a_6
  118. Vector(NA122, NA296) → b_6
  119. Vector(NA122, NA149) → c_6
  120. Vector(NA122, NA150) → d_6
  121. Vector(NA123, NA177) → e_6
  122. Vector(NA123, NA183) → f_6
  123. Vector(NA123, NA126) → g_6
  124. Vector(NA123, NA151) → h_6
  125. Vector(NA123, NA154) → i_6
  126. Vector(NA123, NA129) → j_6
  127. Vector(NA123, NA134) → k_6
  128. Vector(NA123, NA176) → l_6
  129. Vector(NA123, NA191) → m_6
  130. Vector(NA123, NA190) → n_6
  131. Vector(NA125, NA153) → p_6
  132. Vector(NA125, NA126) → q_6
  133. Vector(NA125, NA152) → r_6
  134. Vector(NA126, NA152) → s_6
  135. Vector(NA126, NA133) → t_6
  136. Vector(NA129, NA184) → u_7
  137. Vector(NA129, NA133) → v_7
  138. Vector(NA129, NA130) → w_7
  139. Vector(NA130, MG135) → a_7
  140. Vector(NA130, NA132) → b_7
  141. Vector(NA134, MG159) → c_7
  142. Vector(NA134, NA151) → d_7
  143. Vector(MG135, MG138) → e_7
  144. Vector(MG135, MG136) → f_7
  145. Vector(MG136, MG138) → g_7
  146. Vector(MG136, MG137) → h_7
  147. Vector(MG137, NA150) → i_7
  148. Vector(MG138, MG160) → j_7
  149. Vector(MG138, MG198) → k_7
  150. Vector(NA149, NA182) → l_7
  151. Vector(NA150, NA182) → m_7
  152. Vector(NA150, NA186) → n_7
  153. Vector(NA151, NA183) → p_7
  154. Vector(NA151, NA176) → q_7
  155. Vector(NA151, NA232) → r_7
  156. Vector(NA152, NA153) → s_7
  157. Vector(NA152, NA154) → t_7
  158. Vector(NA152, NA155) → u_8
  159. Vector(NA154, NA184) → v_8
  160. Vector(NA154, SP225) → w_8
  161. Vector(NA133, NA154) → a_8
  162. Vector(NA132, NA154) → b_8
  163. Vector(NA155, NA173) → c_8
  164. Vector(NA155, NA158) → d_8
  165. Vector(NA157, NA173) → e_8
  166. Vector(NA157, NA158) → f_8
  167. Vector(NA158, NA174) → g_8
  168. Vector(NA158, NA187) → h_8
  169. Vector(NA158, NA211) → i_8
  170. Vector(NA173, NA188) → j_8
  171. Vector(NA173, NA193) → k_8
  172. Vector(NA176, NA177) → l_8
  173. Vector(NA176, NA234) → m_8
  174. Vector(NA176, NA296) → n_8
  175. Vector(NA176, MG159) → p_8
  176. Vector(NA176, NA194) → q_8
  177. Vector(NA177, NA179) → r_8
  178. Vector(NA177, NA192) → s_8
  179. Vector(NA177, NA191) → t_8
  180. Vector(NA177, NA190) → u_9
  181. Vector(NA179, NA188) → v_9
  182. Vector(NA179, NA193) → w_9
  183. Vector(NA179, NA180) → a_9
  184. Vector(NA179, NA240) → b_9
  185. Vector(NA179, NA194) → c_9
  186. Vector(MG159, NA234) → d_9
  187. Vector(MG159, MG160) → e_9
  188. Vector(MG159, MG196) → f_9
  189. Vector(MG159, MG216) → g_9
  190. Vector(MG160, MG217) → h_9
  191. Vector(MG160, MG242) → i_9
  192. Vector(MG160, MG218) → j_9
  193. Vector(MG160, MG198) → k_9
  194. Vector(NA182, NA209) → l_9
  195. Vector(NA182, NA239) → m_9
  196. Vector(NA183, NA191) → n_9
  197. Vector(NA184, NA186) → p_9
  198. Vector(NA186, MG197) → q_9
  199. Vector(NA186, NA264) → r_9
  200. Vector(NA188, MG220) → s_9
  201. Vector(NA188, MG221) → t_9
  202. Vector(NA188, NA193) → u_{10}
  203. Vector(NA188, NA211) → v_{10}
  204. Vector(NA188, NA208) → w_{10}
  205. Vector(NA190, NA192) → a_{10}
  206. Vector(NA190, NA213) → b_{10}
  207. Vector(NA191, NA192) → c_{10}
  208. Vector(NA191, NA230) → d_{10}
  209. Vector(NA191, NA232) → e_{10}
  210. Vector(NA192, NA209) → f_{10}
  211. Vector(NA193, NA215) → g_{10}
  212. Vector(NA193, NA296) → h_{10}
  213. Vector(NA193, NA294) → i_{10}
  214. Vector(NA193, NA240) → j_{10}
  215. Vector(NA193, NA194) → k_{10}
  216. Vector(NA193, NA208) → l_{10}
  217. Vector(NA193, NA235) → m_{10}
  218. Vector(NA194, NA234) → n_{10}
  219. Vector(NA194, NA215) → p_{10}
  220. Vector(NA194, NA294) → q_{10}
  221. Vector(NA194, NA208) → r_{10}
  222. Vector(NA194, NA235) → s_{10}
  223. Vector(MG195, MG198) → t_{10}
  224. Vector(MG196, MG216) → u_{11}
  225. Vector(MG197, NA234) → v_{11}
  226. Vector(MG197, MG217) → w_{11}
  227. Vector(MG197, MG216) → a_{11}
  228. Vector(MG197, NA239) → b_{11}
  229. Vector(MG198, NA234) → c_{11}
  230. Vector(NA208, NA294) → d_{11}
  231. Vector(NA209, NA210) → e_{11}
  232. Vector(NA209, NA212) → f_{11}
  233. Vector(NA211, NA229) → g_{11}
  234. Vector(NA212, NA264) → h_{11}
  235. Vector(NA212, NA231) → i_{11}
  236. Vector(NA212, NA265) → j_{11}
  237. Vector(NA213, NA233) → k_{11}
  238. Vector(NA213, NA230) → l_{11}
  239. Vector(NA213, NA232) → m_{11}
  240. Vector(NA294, NA215) → n_{11}
  241. Vector(NA215, NA296) → p_{11}
  242. Vector(MG216, MG217) → q_{11}
  243. Vector(MG216, MG242) → r_{11}
  244. Vector(MG216, MG218) → s_{11}
  245. Vector(MG217, MG242) → t_{11}
  246. Vector(MG218, MG242) → u_{12}
  247. Vector(MG220, MG221) → v_{12}
  248. Vector(MG220, MG223) → w_{12}
  249. Vector(MG221, MG223) → a_{12}
  250. Vector(MG223, MG224) → b_{12}
  251. Vector(MG223, MG273) → c_{12}
  252. Vector(MG224, MG273) → d_{12}
  253. Vector(MG224, MG275) → e_{12}
  254. Vector(SP225, SP246) → f_{12}
  255. Vector(NA230, NA233) → g_{12}
  256. Vector(NA232, NA240) → h_{12}
  257. Vector(NA233, NA241) → i_{12}
  258. Vector(NA233, NA239) → j_{12}
  259. Vector(NA234, NA241) → k_{12}
  260. Vector(NA234, NA235) → l_{12}
  261. Vector(NA239, NA241) → m_{12}
  262. Vector(NA239, NA270) → n_{12}
  263. Vector(NA241, NA266) → p_{12}
  264. Vector(NA241, NA269) → q_{12}
  265. Vector(NA264, NA265) → r_{12}
  266. Segment(K_{28}, M_{28}) → t_{12}
  267. Translate(K_{28}, Vector[(0, 0), (0.8, 0)]) → N_{28}
  268. Translate(M_{28}, Vector[(0, 0), (0.8, 0)]) → O_{28}
  269. Segment(N_{28}, O_{28}) → f_{13}
  270. Translate(K_{28}, Vector[(0, 0), (1.7, 0)]) → P_{28}
  271. Translate(M_{28}, Vector[(0, 0), (1.7, 0)]) → Q_{28}
  272. Segment(P_{28}, Q_{28}) → g_{13}
  273. Translate(K_{28}, Vector[(0, 0), (2.7, 0)]) → R_{28}
  274. Translate(M_{28}, Vector[(0, 0), (2.7, 0)]) → S_{28}
  275. Segment(R_{28}, S_{28}) → h_{13}
  276. Translate(K_{28}, Vector[(0, 0), (3.7, 0)]) → T_{28}
  277. Translate(M_{28}, Vector[(0, 0), (3.7, 0)]) → U_{28}
  278. Segment(T_{28}, U_{28}) → i_{13}
  279. Translate(K_{28}, Vector[(0, 0), (4.8, 0)]) → V_{28}
  280. Translate(M_{28}, Vector[(0, 0), (4.8, 0)]) → W_{28}
  281. Segment(V_{28}, W_{28}) → j_{13}
  282. Translate(K_{28}, Vector[(0, 0), (5.8, 0)]) → Z_{28}
  283. Translate(M_{28}, Vector[(0, 0), (5.8, 0)]) → A_{29}
  284. Segment(Z_{28}, A_{29}) → k_{13}
  285. Translate(K_{28}, Vector[(0, 0), (6.8, 0)]) → B_{29}
  286. Translate(M_{28}, Vector[(0, 0), (6.8, 0)]) → C_{29}
  287. Segment(B_{29}, C_{29}) → l_{13}
  288. Translate(K_{28}, Vector[(0, 0), (7.8, 0)]) → D_{29}
  289. Translate(M_{28}, Vector[(0, 0), (7.8, 0)]) → E_{29}
  290. Segment(D_{29}, E_{29}) → m_{13}
  291. Translate(K_{28}, Vector[(0, 0), (8.8, 0)]) → G_{29}
  292. Translate(M_{28}, Vector[(0, 0), (8.8, 0)]) → H_{29}
  293. Segment(G_{29}, H_{29}) → n_{13}
  294. Translate(K_{28}, Vector[(0, 0), (9.8, 0)]) → I_{29}
  295. Translate(M_{28}, Vector[(0, 0), (9.8, 0)]) → J_{29}
  296. Segment(I_{29}, J_{29}) → p_{13}
  297. Translate(K_{28}, Vector[(0, 0), (10.8, 0)]) → K_{29}
  298. Translate(M_{28}, Vector[(0, 0), (10.8, 0)]) → L_{29}
  299. Segment(K_{29}, L_{29}) → q_{13}
  300. PolyLine(NA003, NA004, NA098, MG115, NA122, MG138, NA149, MG160, NA182, MG198, NA208, SP225, NA012, NA229, SP246, NA264, MG275, NA018, NA026, NA037, NA051, MG062, NA071, MG090) → g
  301. Point(g) → M_{29}

AI 绘图提示语参考

请绘制一个由向量和点组成的几何图形，展示乘法关系。图形包含一系列点、向量和连接线，用于说明乘法原理、百分比变化、随机过程以及定积分的几何解释。构造顺序是从点开始，绘制向量，最后连接成图形，适合高中和早期大学学生使用。
以上提示语为根据原始绘图命令整理而来，仅供参考。