MathPlotAI Logo MathPlotAI

Interactive Vector and Normal Geometry in Coordinate System

Interactive Vector and Normal Geometry in Coordinate System
Loading the math board and drawing, please wait…
2012-11-01 21:34
TrianglePolygonVectorCoordinate GeometryGeometric Construction
This GeoGebra example demonstrates how changing vectors in a coordinate system dynamically updates the configuration of normal vectors and associated geometric shapes, such as triangles and polygons. The core mathematical concepts covered include vector algebra, normal vectors, coordinate geometry (intersection of lines and circles), and geometric transformations including rotation, dilation, midpoint construction, and distance measurement. To build this model, various GeoGebra commands are integrated, such as Line, OrthogonalLine, Circle, Intersect, Polygon, Distance, Rotate, Dilate, Midpoint, If, Segment, and Vector, managing complex objects from coordinates x, y, z to points A through K. It is highly suitable for high school and university students studying linear algebra, vector calculus, and analytical geometry, serving as an excellent visual tool for classroom instruction and interactive exploration.

GeoGebra Commands & Parameters

  1. Line(A, B) → a
  2. OrthogonalLine(C, a) → b
  3. OrthogonalLine(C, b) → c
  4. Circle(C, l) → d
  5. Circle(C, br) → e
  6. Intersect(d, c) → D, E
  7. Intersect(e, b) → F, G
  8. OrthogonalLine(G, b) → f
  9. OrthogonalLine(D, c) → g
  10. OrthogonalLine(F, b) → h
  11. OrthogonalLine(E, c) → i
  12. Intersect(f, i) → H
  13. Intersect(g, f) → I
  14. Intersect(g, h) → J
  15. Intersect(h, i) → K
  16. Polygon(H, I, J, K) → mangekant1, h_1, i_1, j, k
  17. OrthogonalLine(Rotate, h_1) → m
  18. OrthogonalLine(Rotate, k) → n
  19. Intersect(h_1, m) → M
  20. Intersect(k, n) → N
  21. Distance(Rotate, M) → avstandLM
  22. Distance(Rotate, N) → avstandLN
  23. OrthogonalLine(O, a) → p
  24. OrthogonalLine(O, p) → q
  25. Circle(O, size) → r
  26. Intersect(r, q) → P, Q
  27. Intersect(r, p) → R, S
  28. OrthogonalLine(S, p) → s
  29. OrthogonalLine(P, q) → t
  30. OrthogonalLine(R, p) → a_1
  31. OrthogonalLine(Q, q) → b_1
  32. Intersect(s, b_1) → T
  33. Intersect(s, t) → U
  34. Intersect(a_1, t) → V
  35. Intersect(a_1, b_1) → W
  36. Polygon(U, T, W, V) → mangekant2, u, t_1, w, v
  37. Rotate(U, ho, O) → U'
  38. Rotate(T, ho, O) → T'
  39. Rotate(W, ho, O) → W'
  40. Rotate(V, ho, O) → V'
  41. Polygon(U', T', W', V') → mangekant2', u', t', w', v'
  42. OrthogonalLine(T', q) → c_1
  43. OrthogonalLine(W', q) → d_1
  44. OrthogonalLine(V', q) → e_1
  45. OrthogonalLine(U', q) → f_1
  46. OrthogonalLine(T', q) → g_1
  47. OrthogonalLine(V', q) → j_1
  48. OrthogonalLine(U', q) → k_1
  49. OrthogonalLine(W', q) → l_1
  50. Intersect(g_1, q) → Z
  51. Intersect(l_1, q) → A_1
  52. Intersect(q, k_1) → B_1
  53. Intersect(q, j_1) → C_1
  54. Distance(Z, T') → avstandZT'
  55. Distance(B_1, U') → avstandB1U'
  56. Dilate(T', l1, Z) → T''
  57. Dilate(W', l1, A_1) → W''
  58. Dilate(U', l1, B_1) → U''
  59. Dilate(V', l1, C_1) → V''
  60. Polygon(T'', W'', V'', U'') → mangekant3, t'', w'', v'', u''
  61. Midpoint(W'', V'') → D_1
  62. Midpoint(V'', U'') → E_1
  63. Distance(O, D_1) → avstandOD1
  64. Distance(O, E_1) → avstandOE1
  65. If(((-π < ve) ∧ (ve < (-π) / 2) ∨ (π / 2 < ve) ∧ (ve < π)) ∧ ((-π) / 2 < ho) ∧ (ho < π / 2) ∨ ((-π) / 2 < ve) ∧ (ve < π / 2) ∧ ((-π < ho) ∧ (ho < (-π) / 2) ∨ (π / 2 < ho) ∧ (ho < π)), van / (size - xp) + 3, (-van) / (size - xp) - 3) → ix
  66. Dilate(D_1, ix, O) → F_1
  67. If((0 < ho) ∧ (ho < π) ∧ ((-π) / 2 < ve) ∧ (ve < π / 2) ∨ (-π < ho) ∧ (ho < 0) ∧ ((-π < ve) ∧ (ve < (-π) / 2) ∨ (π / 2 < ve) ∧ (ve < π)), van / (size - yp) + 3, (-van) / (size - yp) - 3) → iy
  68. Dilate(E_1, iy, O) → G_1
  69. If((-π < ve) ∧ (ve < (-π) / 2) ∨ (0 < ve) ∧ (ve < π / 2), (-van) / (size - zp) - 3, van / (size - zp) + 3) → iz
  70. Circle(O, size) → p_1
  71. Circle(O, zp) → q_1
  72. Intersect(q_1, p) → J_1, K_1
  73. Dilate(K_1, iz, O) → H_1
  74. Dilate(D_1, xpos, O) → I_1
  75. Dilate(D_1, xneg, O) → L_1
  76. Dilate(E_1, ypos, O) → M_1
  77. Dilate(E_1, yneg, O) → N_1
  78. Line(M_1, F_1) → m_1
  79. Line(N_1, F_1) → n_1
  80. Line(L_1, G_1) → r_1
  81. Line(I_1, G_1) → s_1
  82. Intersect(n_1, r_1) → O_1
  83. Intersect(m_1, r_1) → P_1
  84. Intersect(s_1, m_1) → Q_1
  85. Intersect(s_1, n_1) → R_1
  86. Polygon(P_1, O_1, R_1, Q_1) → mangekant4, p_2, o_1, r_2, q_2
  87. Dilate(K_1, zpos, O) → S_1
  88. Dilate(K_1, zneg, O) → T_1
  89. Segment(I_1, L_1) → a_2
  90. Segment(N_1, M_1) → b_2
  91. Line(F_1, S_1) → c_2
  92. Line(T_1, F_1) → d_2
  93. Line(S_1, G_1) → e_2
  94. Line(T_1, G_1) → f_2
  95. Line(H_1, N_1) → g_2
  96. Line(H_1, I_1) → h_2
  97. Line(H_1, L_1) → i_2
  98. Line(H_1, M_1) → j_2
  99. Intersect(g_2, e_2) → V_1
  100. Intersect(g_2, f_2) → W_1
  101. Intersect(e_2, j_2) → Z_1
  102. Intersect(f_2, j_2) → A_2
  103. Line(V_1, F_1) → k_2
  104. Line(Z_1, F_1) → l_2
  105. Line(W_1, F_1) → m_2
  106. Line(A_2, F_1) → n_2
  107. Intersect(i_2, c_2) → B_2
  108. Intersect(i_2, d_2) → C_2
  109. Intersect(c_2, h_2) → D_2
  110. Intersect(h_2, d_2) → E_2
  111. Line(B_2, G_1) → s_2
  112. Line(D_2, G_1) → t_2
  113. Line(E_2, G_1) → a_3
  114. Line(C_2, G_1) → b_3
  115. Intersect(k_2, s_2) → F_2
  116. Intersect(s_2, l_2) → G_2
  117. Intersect(t_2, l_2) → H_2
  118. Intersect(t_2, k_2) → I_2
  119. Intersect(m_2, b_3) → J_2
  120. Intersect(m_2, a_3) → K_2
  121. Intersect(a_3, n_2) → L_2
  122. Intersect(n_2, b_3) → M_2
  123. Segment(F_2, G_2) → c_3
  124. Segment(G_2, H_2) → d_3
  125. Segment(H_2, I_2) → e_3
  126. Segment(I_2, F_2) → f_3
  127. Segment(F_2, J_2) → g_3
  128. Segment(I_2, K_2) → h_3
  129. Segment(L_2, H_2) → i_3
  130. Segment(G_2, M_2) → j_3
  131. Segment(M_2, L_2) → k_3
  132. Segment(L_2, K_2) → l_3
  133. Segment(K_2, J_2) → m_3
  134. Segment(J_2, M_2) → n_3
  135. Segment(T_1, S_1) → p_3
  136. Dilate(I_1, enx, O) → N_2
  137. Dilate(I_1, tox, O) → O_2
  138. Dilate(I_1, trex, O) → P_2
  139. Dilate(I_1, fix, O) → Q_2
  140. Dilate(I_1, fex, O) → R_2
  141. Dilate(I_1, enxm, O) → S_2
  142. Dilate(I_1, six, O) → T_2
  143. Dilate(I_1, sevx, O) → U_2
  144. Dilate(I_1, eix, O) → V_2
  145. Dilate(I_1, nix, O) → W_2
  146. Dilate(I_1, tex, O) → Z_2
  147. Dilate(I_1, toxm, O) → A_3
  148. Dilate(I_1, trexm, O) → B_3
  149. Dilate(I_1, fixm, O) → C_3
  150. Dilate(I_1, fexm, O) → D_3
  151. Dilate(I_1, sixm, O) → E_3
  152. Dilate(I_1, sevxm, O) → F_3
  153. Dilate(I_1, eixm, O) → G_3
  154. Dilate(I_1, nixm, O) → H_3
  155. Dilate(I_1, texm, O) → I_3
  156. Line(Z_2, G_1) → q_3
  157. Line(W_2, G_1) → r_3
  158. Line(V_2, G_1) → s_3
  159. Line(U_2, G_1) → t_3
  160. Line(T_2, G_1) → a_4
  161. Line(R_2, G_1) → b_4
  162. Line(Q_2, G_1) → c_4
  163. Line(P_2, G_1) → d_4
  164. Line(O_2, G_1) → e_4
  165. Line(N_2, G_1) → f_4
  166. Line(S_2, G_1) → g_4
  167. Line(A_3, G_1) → h_4
  168. Line(B_3, G_1) → i_4
  169. Line(C_3, G_1) → j_4
  170. Line(D_3, G_1) → k_4
  171. Line(E_3, G_1) → l_4
  172. Line(F_3, G_1) → m_4
  173. Line(G_3, G_1) → n_4
  174. Line(H_3, G_1) → p_4
  175. Line(I_3, G_1) → q_4
  176. Intersect(q_2, q_4) → J_3
  177. Intersect(q_2, p_4) → K_3
  178. Intersect(q_2, n_4) → L_3
  179. Intersect(q_2, m_4) → M_3
  180. Intersect(q_2, l_4) → N_3
  181. Intersect(q_2, k_4) → O_3
  182. Intersect(q_2, j_4) → P_3
  183. Intersect(q_2, i_4) → Q_3
  184. Intersect(q_2, h_4) → R_3
  185. Intersect(q_2, g_4) → S_3
  186. Intersect(q_2, f_4) → T_3
  187. Intersect(q_2, e_4) → U_3
  188. Intersect(q_2, d_4) → V_3
  189. Intersect(q_2, c_4) → W_3
  190. Intersect(q_2, b_4) → Z_3
  191. Intersect(q_2, a_4) → A_4
  192. Intersect(q_2, t_3) → B_4
  193. Intersect(q_2, s_3) → C_4
  194. Intersect(q_2, r_3) → D_4
  195. Intersect(q_2, q_3) → E_4
  196. Intersect(o_1, q_3) → F_4
  197. Intersect(o_1, r_3) → G_4
  198. Intersect(o_1, s_3) → H_4
  199. Intersect(o_1, t_3) → I_4
  200. Intersect(o_1, a_4) → J_4
  201. Intersect(o_1, b_4) → K_4
  202. Intersect(o_1, c_4) → L_4
  203. Intersect(o_1, d_4) → M_4
  204. Intersect(o_1, e_4) → N_4
  205. Intersect(o_1, f_4) → O_4
  206. Intersect(o_1, g_4) → P_4
  207. Intersect(o_1, h_4) → Q_4
  208. Intersect(o_1, i_4) → R_4
  209. Intersect(o_1, j_4) → S_4
  210. Intersect(o_1, k_4) → T_4
  211. Intersect(o_1, m_4) → W_4
  212. Intersect(o_1, n_4) → Z_4
  213. Intersect(o_1, p_4) → A_5
  214. Intersect(o_1, q_4) → B_5
  215. Segment(B_5, J_3) → r_4
  216. Segment(A_5, K_3) → s_4
  217. Segment(Z_4, L_3) → t_4
  218. Segment(W_4, M_3) → a_5
  219. Segment(T_4, O_3) → c_5
  220. Segment(S_4, P_3) → d_5
  221. Segment(R_4, Q_3) → e_5
  222. Segment(Q_4, R_3) → f_5
  223. Segment(P_4, S_3) → g_5
  224. Segment(O_4, T_3) → h_5
  225. Segment(N_4, U_3) → i_5
  226. Segment(M_4, V_3) → j_5
  227. Segment(L_4, W_3) → k_5
  228. Segment(K_4, Z_3) → l_5
  229. Segment(J_4, A_4) → m_5
  230. Segment(I_4, B_4) → n_5
  231. Segment(H_4, C_4) → p_5
  232. Segment(G_4, D_4) → q_5
  233. Segment(F_4, E_4) → r_5
  234. Dilate(M_1, eny, O) → C_5
  235. Dilate(M_1, toy, O) → D_5
  236. Dilate(M_1, trey, O) → E_5
  237. Dilate(M_1, firy, O) → F_5
  238. Dilate(M_1, femy, O) → G_5
  239. Dilate(M_1, sixy, O) → H_5
  240. Dilate(M_1, sevy, O) → I_5
  241. Dilate(M_1, eiy, O) → J_5
  242. Dilate(M_1, niy, O) → K_5
  243. Dilate(M_1, tey, O) → L_5
  244. Dilate(M_1, toym, O) → M_5
  245. Dilate(M_1, enym, O) → N_5
  246. Dilate(M_1, treym, O) → O_5
  247. Dilate(M_1, firym, O) → P_5
  248. Dilate(M_1, sixym, O) → Q_5
  249. Dilate(M_1, femym, O) → R_5
  250. Dilate(M_1, sevym, O) → S_5
  251. Dilate(M_1, eiym, O) → T_5
  252. Dilate(M_1, niym, O) → U_5
  253. Dilate(M_1, teym, O) → V_5
  254. Line(V_5, F_1) → s_5
  255. Line(U_5, F_1) → t_5
  256. Line(T_5, F_1) → a_6
  257. Line(S_5, F_1) → b_6
  258. Line(Q_5, F_1) → c_6
  259. Line(R_5, F_1) → d_6
  260. Line(P_5, F_1) → e_6
  261. Line(O_5, F_1) → f_6
  262. Line(M_5, F_1) → g_6
  263. Line(N_5, F_1) → h_6
  264. Line(C_5, F_1) → i_6
  265. Line(D_5, F_1) → j_6
  266. Line(E_5, F_1) → k_6
  267. Line(F_5, F_1) → l_6
  268. Line(G_5, F_1) → m_6
  269. Line(H_5, F_1) → n_6
  270. Line(I_5, F_1) → p_6
  271. Line(J_5, F_1) → q_6
  272. Line(K_5, F_1) → r_6
  273. Line(L_5, F_1) → s_6
  274. Intersect(p_2, s_5) → W_5
  275. Intersect(p_2, t_5) → Z_5
  276. Intersect(p_2, a_6) → A_6
  277. Intersect(p_2, b_6) → B_6
  278. Intersect(p_2, c_6) → C_6
  279. Intersect(p_2, d_6) → D_6
  280. Intersect(p_2, e_6) → E_6
  281. Intersect(p_2, f_6) → F_6
  282. Intersect(p_2, g_6) → G_6
  283. Intersect(p_2, h_6) → H_6
  284. Intersect(p_2, i_6) → I_6
  285. Intersect(p_2, j_6) → J_6
  286. Intersect(p_2, k_6) → K_6
  287. Intersect(p_2, l_6) → L_6
  288. Intersect(p_2, m_6) → M_6
  289. Intersect(p_2, n_6) → N_6
  290. Intersect(p_2, p_6) → O_6
  291. Intersect(p_2, q_6) → P_6
  292. Intersect(p_2, r_6) → Q_6
  293. Intersect(p_2, s_6) → R_6
  294. Intersect(r_2, s_6) → S_6
  295. Intersect(r_2, r_6) → T_6
  296. Intersect(r_2, q_6) → U_6
  297. Intersect(r_2, p_6) → V_6
  298. Intersect(r_2, n_6) → W_6
  299. Intersect(r_2, m_6) → Z_6
  300. Intersect(r_2, l_6) → A_7
  301. Intersect(r_2, k_6) → B_7
  302. Intersect(r_2, j_6) → C_7
  303. Intersect(r_2, i_6) → D_7
  304. Intersect(r_2, h_6) → E_7
  305. Intersect(r_2, g_6) → F_7
  306. Intersect(r_2, f_6) → G_7
  307. Intersect(r_2, e_6) → H_7
  308. Intersect(r_2, d_6) → I_7
  309. Intersect(r_2, c_6) → J_7
  310. Intersect(r_2, b_6) → K_7
  311. Intersect(r_2, a_6) → L_7
  312. Intersect(r_2, t_5) → M_7
  313. Intersect(r_2, s_5) → N_7
  314. Segment(N_7, W_5) → t_6
  315. Segment(M_7, Z_5) → a_7
  316. Segment(L_7, A_6) → b_7
  317. Segment(K_7, B_6) → c_7
  318. Segment(J_7, C_6) → d_7
  319. Segment(I_7, D_6) → e_7
  320. Segment(H_7, E_6) → f_7
  321. Segment(G_7, F_6) → g_7
  322. Segment(F_7, G_6) → h_7
  323. Segment(E_7, H_6) → i_7
  324. Segment(D_7, I_6) → j_7
  325. Segment(C_7, J_6) → k_7
  326. Segment(B_7, K_6) → l_7
  327. Segment(A_7, L_6) → m_7
  328. Segment(Z_6, M_6) → n_7
  329. Segment(W_6, N_6) → p_7
  330. Segment(V_6, O_6) → q_7
  331. Segment(U_6, P_6) → r_7
  332. Segment(T_6, Q_6) → s_7
  333. Segment(S_6, R_6) → t_7
  334. Intersect(o_1, l_4) → U_4
  335. Intersect(q_2, l_4) → V_4
  336. Segment(U_4, V_4) → b_5
  337. Dilate(S_1, enz, O) → O_7
  338. Vector(O, N_2) → z
  339. Vector(O, C_5) → u_1
  340. Vector(O, O_7) → v_1
  341. Dilate(I_1, xx1, O) → P_7
  342. Dilate(M_1, yy1, O) → Q_7
  343. Dilate(S_1, zz1, O) → R_7
  344. Line(P_7, G_1) → a_8
  345. Line(R_7, G_1) → b_8
  346. Line(Q_7, F_1) → c_8
  347. Line(R_7, F_1) → d_8
  348. Line(H_1, P_7) → e_8
  349. Line(H_1, Q_7) → f_8
  350. Intersect(b_8, f_8) → S_7
  351. Intersect(e_8, d_8) → T_7
  352. Line(T_7, G_1) → g_8
  353. Line(S_7, F_1) → h_8
  354. Intersect(h_8, g_8) → P1
  355. Intersect(c_8, a_8) → V_7
  356. Segment(P1, R_7) → i_8
  357. Segment(P1, V_7) → j_8
  358. Segment(V_7, P_7) → k_8
  359. Segment(V_7, Q_7) → l_8
  360. Dilate(I_1, xx2, O) → L
  361. Dilate(M_1, yy2, O) → W_7
  362. Dilate(S_1, zz2, O) → Z_7
  363. Line(H_1, L) → m_8
  364. Line(H_1, W_7) → n_8
  365. Line(W_7, F_1) → p_8
  366. Line(Z_7, F_1) → q_8
  367. Line(L, G_1) → r_8
  368. Line(Z_7, G_1) → s_8
  369. Intersect(m_8, q_8) → A_8
  370. Intersect(n_8, s_8) → B_8
  371. Line(A_8, G_1) → t_8
  372. Line(B_8, F_1) → a_9
  373. Intersect(t_8, a_9) → P2
  374. Intersect(p_8, r_8) → D_8
  375. Segment(W_7, D_8) → b_9
  376. Segment(D_8, L) → c_9
  377. Segment(D_8, P2) → d_9
  378. Segment(P2, Z_7) → e_9
  379. Dilate(I_1, xx3, O) → E_8
  380. Dilate(M_1, yy3, O) → F_8
  381. Dilate(S_1, zz3, O) → G_8
  382. Line(H_1, F_8) → f_9
  383. Line(H_1, E_8) → g_9
  384. Line(F_8, F_1) → h_9
  385. Line(G_8, F_1) → i_9
  386. Line(E_8, G_1) → j_9
  387. Line(G_8, G_1) → k_9
  388. Intersect(g_9, i_9) → H_8
  389. Intersect(f_9, k_9) → I_8
  390. Line(H_8, G_1) → l_9
  391. Line(I_8, F_1) → m_9
  392. Intersect(m_9, l_9) → P3
  393. Intersect(j_9, h_9) → K_8
  394. Segment(F_8, K_8) → n_9
  395. Segment(K_8, P3) → p_9
  396. Segment(K_8, E_8) → q_9
  397. Segment(P3, G_8) → r_9
  398. Vector(P1, P2) → w_1
  399. Vector(P1, P3) → z_1
  400. Dilate(I_1, nnx, O) → U_7
  401. Dilate(M_1, nny, O) → C_8
  402. Dilate(S_1, nnz, O) → J_8
  403. Line(H_1, U_7) → s_9
  404. Line(H_1, C_8) → t_9
  405. Line(C_8, F_1) → a_{10}
  406. Line(J_8, F_1) → b_{10}
  407. Line(U_7, G_1) → c_{10}
  408. Line(J_8, G_1) → d_{10}
  409. Intersect(s_9, b_{10}) → L_8
  410. Intersect(t_9, d_{10}) → M_8
  411. Line(L_8, G_1) → e_{10}
  412. Line(M_8, F_1) → f_{10}
  413. Intersect(e_{10}, f_{10}) → N_8
  414. Intersect(c_{10}, a_{10}) → O_8
  415. Vector(P1, N_8) → u_2

AI Drawing Prompt Reference

Demonstrate dynamic vector and normal vector geometry in a coordinate system, with associated shapes like triangles and polygons. Adjust vectors to see how normal vectors, lines, circles, and intersections change, applying transformations such as rotation, dilation, midpoint construction, and distance measurement to build complex figures.
The prompt above was compiled from the original drawing commands and is for reference only.