几何方式相加向量：平行移位与首尾相接

几何画板及绘图内容加载中，请稍等……

2020-09-11 03:50

本范例用交互式图形展示“几何相加向量”这一基本法则：把向量在平面中平移并进行首尾相接，其合向量对应的方向与大小会在图中清晰形成。核心数学概念包括二维向量、向量的几何加法（首尾相接法/平行四边形法）、向量平移（translate）以及两对象的交点判定（intersect）以确定构造出的关键点。若配合测验式界面，还可引导学生理解角度、线性关系与最优化/检验类任务中“合向量是否匹配”的判断思路。作图与构造要点（由构造摘要提炼）：使用 Vector 创建相关向量；用 Translate 对向量或几何对象进行平移以形成“首尾相接”的几何关系；通过 Intersect 计算由线段/射线等生成的交点位置，作为合向量端点或中间约束点；用 Sequence 生成按步骤迭代或串联的点/对象集合；用 RandomElement 引入随机元素以生成不同题目或扰动；对象如由点 A、B、C、D、E、F、G 及其变换后的 A'、E' 等参与几何布局，并以 button1、text1、text2 组织交互与说明。适合学段：中学数学（向量入门到二维解析几何的关联阶段），也适用于大学预科或入门课程的可视化练习与课堂测验使用。

GeoGebra 命令及参数

Intersect(xAxis, yAxis) → B
Sequence(Sequence[(i, j), i, -10, 10, 1], j, -10, 10, 1) → l2, {{(-10, -10), (-9, -10), (-8, -10), (-7, -10), (-6, -10), (-5, -10), (-4, -10), (-3, -10), (-2, -10), (-1, -10), (0, -10), (1, -10), (2, -10), (3, -10), (4, -10), (5, -10), (6, -10), (7, -10), (8, -10), (9, -10), (10, -10)}, {(-10, -9), (-9, -9), (-8, -9), (-7, -9), (-6, -9), (-5, -9), (-4, -9), (-3, -9), (-2, -9), (-1, -9), (0, -9), (1, -9), (2, -9), (3, -9), (4, -9), (5, -9), (6, -9), (7, -9), (8, -9), (9, -9), (10, -9)}, {(-10, -8), (-9, -8), (-8, -8), (-7, -8), (-6, -8), (-5, -8), (-4, -8), (-3, -8), (-2, -8), (-1, -8), (0, -8), (1, -8), (2, -8), (3, -8), (4, -8), (5, -8), (6, -8), (7, -8), (8, -8), (9, -8), (10, -8)}, {(-10, -7), (-9, -7), (-8, -7), (-7, -7), (-6, -7), (-5, -7), (-4, -7), (-3, -7), (-2, -7), (-1, -7), (0, -7), (1, -7), (2, -7), (3, -7), (4, -7), (5, -7), (6, -7), (7, -7), (8, -7), (9, -7), (10, -7)}, {(-10, -6), (-9, -6), (-8, -6), (-7, -6), (-6, -6), (-5, -6), (-4, -6), (-3, -6), (-2, -6), (-1, -6), (0, -6), (1, -6), (2, -6), (3, -6), (4, -6), (5, -6), (6, -6), (7, -6), (8, -6), (9, -6), (10, -6)}, {(-10, -5), (-9, -5), (-8, -5), (-7, -5), (-6, -5), (-5, -5), (-4, -5), (-3, -5), (-2, -5), (-1, -5), (0, -5), (1, -5), (2, -5), (3, -5), (4, -5), (5, -5), (6, -5), (7, -5), (8, -5), (9, -5), (10, -5)}, {(-10, -4), (-9, -4), (-8, -4), (-7, -4), (-6, -4), (-5, -4), (-4, -4), (-3, -4), (-2, -4), (-1, -4), (0, -4), (1, -4), (2, -4), (3, -4), (4, -4), (5, -4), (6, -4), (7, -4), (8, -4), (9, -4), (10, -4)}, {(-10, -3), (-9, -3), (-8, -3), (-7, -3), (-6, -3), (-5, -3), (-4, -3), (-3, -3), (-2, -3), (-1, -3), (0, -3), (1, -3), (2, -3), (3, -3), (4, -3), (5, -3), (6, -3), (7, -3), (8, -3), (9, -3), (10, -3)}, {(-10, -2), (-9, -2), (-8, -2), (-7, -2), (-6, -2), (-5, -2), (-4, -2), (-3, -2), (-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (3, -2), (4, -2), (5, -2), (6, -2), (7, -2), (8, -2), (9, -2), (10, -2)}, {(-10, -1), (-9, -1), (-8, -1), (-7, -1), (-6, -1), (-5, -1), (-4, -1), (-3, -1), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1), (3, -1), (4, -1), (5, -1), (6, -1), (7, -1), (8, -1), (9, -1), (10, -1)}, {(-10, 0), (-9, 0), (-8, 0), (-7, 0), (-6, 0), (-5, 0), (-4, 0), (-3, 0), (-2, 0), (-1, 0), (0, 0), (1, 0), (2, 0), (3, 0), (4, 0), (5, 0), (6, 0), (7, 0), (8, 0), (9, 0), (10, 0)}, {(-10, 1), (-9, 1), (-8, 1), (-7, 1), (-6, 1), (-5, 1), (-4, 1), (-3, 1), (-2, 1), (-1, 1), (0, 1), (1, 1), (2, 1), (3, 1), (4, 1), (5, 1), (6, 1), (7, 1), (8, 1), (9, 1), (10, 1)}, {(-10, 2), (-9, 2), (-8, 2), (-7, 2), (-6, 2), (-5, 2), (-4, 2), (-3, 2), (-2, 2), (-1, 2), (0, 2), (1, 2), (2, 2), (3, 2), (4, 2), (5, 2), (6, 2), (7, 2), (8, 2), (9, 2), (10, 2)}, {(-10, 3), (-9, 3), (-8, 3), (-7, 3), (-6, 3), (-5, 3), (-4, 3), (-3, 3), (-2, 3), (-1, 3), (0, 3), (1, 3), (2, 3), (3, 3), (4, 3), (5, 3), (6, 3), (7, 3), (8, 3), (9, 3), (10, 3)}, {(-10, 4), (-9, 4), (-8, 4), (-7, 4), (-6, 4), (-5, 4), (-4, 4), (-3, 4), (-2, 4), (-1, 4), (0, 4), (1, 4), (2, 4), (3, 4), (4, 4), (5, 4), (6, 4), (7, 4), (8, 4), (9, 4), (10, 4)}, {(-10, 5), (-9, 5), (-8, 5), (-7, 5), (-6, 5), (-5, 5), (-4, 5), (-3, 5), (-2, 5), (-1, 5), (0, 5), (1, 5), (2, 5), (3, 5), (4, 5), (5, 5), (6, 5), (7, 5), (8, 5), (9, 5), (10, 5)}, {(-10, 6), (-9, 6), (-8, 6), (-7, 6), (-6, 6), (-5, 6), (-4, 6), (-3, 6), (-2, 6), (-1, 6), (0, 6), (1, 6), (2, 6), (3, 6), (4, 6), (5, 6), (6, 6), (7, 6), (8, 6), (9, 6), (10, 6)}, {(-10, 7), (-9, 7), (-8, 7), (-7, 7), (-6, 7), (-5, 7), (-4, 7), (-3, 7), (-2, 7), (-1, 7), (0, 7), (1, 7), (2, 7), (3, 7), (4, 7), (5, 7), (6, 7), (7, 7), (8, 7), (9, 7), (10, 7)}, {(-10, 8), (-9, 8), (-8, 8), (-7, 8), (-6, 8), (-5, 8), (-4, 8), (-3, 8), (-2, 8), (-1, 8), (0, 8), (1, 8), (2, 8), (3, 8), (4, 8), (5, 8), (6, 8), (7, 8), (8, 8), (9, 8), (10, 8)}, {(-10, 9), (-9, 9), (-8, 9), (-7, 9), (-6, 9), (-5, 9), (-4, 9), (-3, 9), (-2, 9), (-1, 9), (0, 9), (1, 9), (2, 9), (3, 9), (4, 9), (5, 9), (6, 9), (7, 9), (8, 9), (9, 9), (10, 9)}, {(-10, 10), (-9, 10), (-8, 10), (-7, 10), (-6, 10), (-5, 10), (-4, 10), (-3, 10), (-2, 10), (-1, 10), (0, 10), (1, 10), (2, 10), (3, 10), (4, 10), (5, 10), (6, 10), (7, 10), (8, 10), (9, 10), (10, 10)}}
RandomElement(RandomElement[l2]) → C
RandomElement(RandomElement[l2]) → D
Vector(B, D) → u
Vector(B, C) → v
Translate(A, v) → A'
Vector(A, A') → w
Translate(E, u) → E'
Vector(E, E') → a
Vector(F, G) → b

AI 绘图提示语参考

请绘制一个展示二维向量几何加法的图形。图形中包含点A、B、C、D、E、F、G以及它们的平移点A'和E'。首先确定B为x轴与y轴的交点，然后从B随机选取两个点C和D。接着画出向量u从B指向D，向量v从B指向C。将点A平移得到A'，并画出向量w从A指向A'。将点E平移得到E'，并画出向量a从E指向E'。同时画出向量b从F指向G。最后通过平移和连接这些向量，展示向量加法的几何意义：向量u与v的平移后端点相连，形成结果向量。图形应清晰显示各向量的方向、长度以及平移后的位置关系。

以上提示语为根据原始绘图命令整理而来，仅供参考。