高中数学必修第二册“向量的物理背景与概念”教学设计

高中数学必修第二册“向量的物理背景与概念”教学设计  一、教学内容解析  【基础】【核心概念】本节课“向量的物理背景与概念”是高中数学必修第二册平面向量一章的起始课。向量作为近代数学中重要且基本的概念之一，是沟通代数、几何与三角函数的一种工具，有着极其丰富的物理背景和广泛的实际应用。本节课的核心内容是从位移、速度、力等物理量中抽象出向量的概念，理解向量的几何表示，掌握向量的模、零向量、单位向量、平行向量、相等向量等基本概念。这些概念是后续学习向量运算、向量共线定理、平面向量基本定理以及空间向量的基石，对培养学生的数学抽象、直观想象和逻辑推理素养具有不可替代的作用。  【重要】教材编排遵循从具体到抽象、从特殊到一般的认知规律。首先，通过“老鼠由A向西北逃窜，猫由B向正东追去，猫能抓到老鼠吗？”等生活化、趣味性的问题情境，引导学生关注物理中的“位移”、“速度”、“力”这些既有大小又有方向的量，激发认知冲突，从而抽象出向量的本质特征——既有大小又有方向。接着，引入有向线段这一直观工具来表示向量，使抽象的向量概念获得几何直观的支撑。在此基础上，逐步定义向量的模（大小）、零向量、单位向量，并通过方向与大小两个维度对向量进行分类和辨析，引入平行向量（共线向量）与相等向量的概念。整个设计逻辑清晰，层次分明，旨在帮助学生构建清晰、系统的向量知识结构。  【难点】本节课的难点在于对“平行向量”与“相等向量”概念的理解，特别是对向量平行与有向线段所在直线平行的关系辨析，以及如何理解“任意一组平行向量都可以平移到同一条直线上”这一特性。学生受已有知识（如线段平行、直线平行）的干扰，容易将向量的平行与几何中的平行概念混淆，忽视向量平行的本质是方向相同或相反，而与起点位置无关。此外，对零向量方向的任意性及其与平行向量关系的理解，也是学生容易产生困惑的地方。  二、教学目标设计  （一）知识与技能目标  1.【基础】理解向量的概念，掌握向量的几何表示，能区分向量与有向线段。  2.【基础】理解向量的模、零向量、单位向量的概念，并会进行简单的求解与判断。  3.【重要】理解平行向量、相等向量的概念，能在图形中识别并作出符合条件的向量。  （二）过程与方法目标  1.通过观察、分析、类比物理中的位移、速度、力等实例，经历从实际问题中抽象出数学模型的过程，体会数学抽象的思想方法。  2.通过对向量概念及其关系的辨析与讨论，培养学生从多角度、多层次分析问题的能力，提升思维的严谨性。  3.通过运用有向线段表示向量，引导学生感悟数形结合的思想方法，初步建立用向量观点看世界的意识。  （三）情感、态度与价值观目标  1.感受数学与物理、生活的密切联系，激发学生学习数学的兴趣和探索未知的热情。  2.在概念的辨析与讨论中，培养学生敢于质疑、善于思考、严谨求实的科学态度。  3.通过合作学习，培养学生的团队协作精神和勇于表达的交流意识。  三、教学重点与难点  【重点】  1.向量的概念及其几何表示。  2.零向量、单位向量、平行向量、相等向量的概念。  【难点】  1.平行向量与相等向量的概念辨析及其关系。  2.对零向量方向任意性的理解。  四、教学策略与学法指导  （一）教学策略  本节课采用“问题情境—抽象概括—概念形成—辨析应用—反思建构”的教学模式。教师作为学习的组织者、引导者和合作者，通过创设生动、直观的物理情境和生活实例，引发学生的认知需求，引导学生在观察、思考、讨论中自主建构向量的概念体系。教学中注重启发式与探究式相结合，充分利用多媒体课件（如动画演示位移、速度的合成与分解）辅助教学，帮助学生突破难点，建立直观印象。  （二）学法指导  指导学生运用“类比迁移、自主探究、合作交流”的方法进行学习。引导学生将物理中“矢量”的概念类比迁移到数学中，通过小组讨论、师生互动等形式，对向量的关键概念进行多角度辨析。鼓励学生动手画图，在纸上用有向线段表示各种向量，通过直观操作加深对概念的理解，实现从“听数学”到“做数学”的转变。  五、教学实施过程  （一）创设情境，引入新知（约5分钟）  教师活动：利用多媒体展示两幅动态情境。  情境一：展示一只蚂蚁从A点出发，沿直线爬到距A点2厘米的B点。再展示另一只蚂蚁从同一点A出发，先向东爬行1厘米，再向北爬行1厘米，最终到达C点。  提问：两只蚂蚁的爬行路径长度相同吗？它们最终的位置相同吗？要唯一确定蚂蚁的最终位置，除了知道爬行的距离，还需要知道什么？  学生观察、思考、回答：需要知道方向。  情境二：播放一段高铁从北京开往上海的动画，屏幕上显示“时速300公里，由北京向上海方向行驶”。同时，展示一只鸽子在空中匀速直线飞行。  提问：描述高铁和鸽子的运动状态，通常需要哪些量？这些量与我们之前学过的长度、质量等量有什么本质区别？  学生讨论，初步意识到有些量（如速度、位移）不仅有“大小”，还有“方向”。  教师顺势引导：在物理中，我们把既有大小又有方向的量称为矢量。在数学中，我们也有一个专门的概念来描述这类量，这就是我们今天要学习的——向量。  （二）抽象概念，形成定义（约8分钟）  【重要】教师引导学生回顾物理中的具体实例：位移、速度、力。  1.位移：物体从起点到终点的有向线段。它不仅表示移动的距离，还表示移动的方向。例如，从A到B的位移和从B到A的位移是不同的。  2.速度：物体运动的快慢程度（速率）和运动方向。例如，汽车以60km/h的速度向东行驶和向西行驶，速度不同。  3.力：力是物体间的相互作用，不仅有大小，还有作用方向。例如，提着重物时，手对物体施加一个向上的力。  教师总结：这些物理量共同的特征是“既有大小，又有方向”。在数学上，我们撇开它们的具体物理属性，抽象出“向量”的概念。  板书定义：  【基础】既有大小，又有方向的量称为向量（vector）。  教师强调：向量是“量”，是一个数学概念，而物理中的位移、速度、力是向量的具体背景或物理模型。  为了加深理解，教师提出问题，引导学生辨析：  问题1：下列哪些量是向量？  （1）某人身高175cm；（2）从A地到B地的路程是5km；（3）从A地到B地的位移是5km，方向东偏北30°；（4）物体的质量是10kg；（5）物体受到的重力是10N，方向竖直向下；（6）某日气温为25℃。  学生抢答，教师点评，巩固对向量“双重属性”（大小和方向）的认识。  （三）几何表示，深化理解（约10分钟）  【重要】教师指出：为了直观地研究向量，我们需要一种几何表示方法。数学上，我们常用有向线段来表示向量。  1.有向线段：规定了起点和终点的线段。记作AB→\overrightarrow{AB}AB<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">，其中A是起点，B是终点。线段的长度表示向量的大小，箭头所指的方向表示向量的方向。  2.向量的表示方法：  （1）几何表示：用有向线段表示，如AB→\overrightarrow{AB}AB<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">，CD→\overrightarrow{CD}CD<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">。  （2）字母表示：用黑体小写字母表示，如a\boldsymbol{a}a,b\boldsymbol{b}b,c\boldsymbol{c}c（书写时常用a⃗\vec{a}a<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">,b⃗\vec{b}b<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">,c⃗\vec{c}c<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">表示）。  板书示例：画出有向线段AB→\overrightarrow{AB}AB<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">，并在旁边标注a⃗=AB→\vec{a}=\overrightarrow{AB}a<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">=AB<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">。  3.辨析：向量与有向线段的关系。  教师提问：向量是既有大小又有方向的量，有向线段也具有大小和方向。那么，能否说“向量就是有向线段，有向线段就是向量”？  引导学生小组讨论。教师通过几何画板演示：将一个有向线段AB→\overrightarrow{AB}AB<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">平移到A′B′→\overrightarrow{A'B'}A′B′<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">的位置，起点和终点都变了，但它的长度和方向都没有变。  学生观察发现：平移后的有向线段代表的是同一个向量。  教师总结：向量是自由的，只由大小和方向唯一确定，与起点无关。而有向线段是向量的直观几何表示，但它有固定的起点。因此，我们说向量是自由向量，可以用起点不同的有向线段来表示，但这些有向线段是平行的且同向的。这就如同一个人可以站在不同位置，但他还是同一个人。  【难点澄清】向量的平行与有向线段所在直线平行既有联系又有区别。当用有向线段表示向量时，如果两个向量平行（即方向相同或相反），那么表示它们的有向线段所在的直线可能是平行的（如果起点不同），也可能是共线的（如果起点相同或通过平移后可以落在同一直线上）。因此，平行向量也称为共线向量。  （四）概念细化，构建体系（约12分钟）  在理解了向量的基本概念和表示法后，教师引导学生从“大小”和“方向”两个维度对向量进行进一步的分类和定义，逐步完善概念体系。  1.向量的模（长度）  【基础】向量的大小（或长度）称为向量的模。  记作：∣AB→∣|\overrightarrow{AB}|∣AB<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">∣或∣a∣|\boldsymbol{a}|∣a∣。  强调：模是一个非负实数。例如，∣AB→∣=3|\overrightarrow{AB}|=3∣AB<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">∣=3表示该向量的长度为3个单位。  2.两个特殊向量  （1）零向量：长度为0的向量称为零向量。  记作：0\boldsymbol{0}0或0⃗\vec{0}0<pathd="M37720c05.3331..514Sc4.66708.6671.3.3332.6676.667910196.66724.66720.33343..3334.6671110.66711180s6.c28.66714.66753.66735.1.3331.3333.1673.55.56.5s44.83355.5c1.6672.51.3334.52s4.333171c4.66709.1671.83313.55.5S337184337178c012.66715.66732.H213l1711c8..333131904.6674.33311.h359c1625.33324452459z">。  【难点】零向量的方向是任意的，或者说是不确定的。它与任何向量平行（共线）。  教师举例：一个点从位置A移动到位置A（即没有移动），其位移是零向量。  （2）单位向量：长度（模）为1个单位长度的向量称为单位向量。  提问：对于一个非零向量a\boldsymbol{a}a，与它同方向的单位向量该如何表示？它有几个？  引导学生思考并得出：与a\boldsymbol{a}a同方向的单位向量为a∣a∣\frac{\boldsymbol{a}}{|\boldsymbol{a}|}∣a∣a​，它是唯一的。  3.向量间的关系  （1）平行向量（共线向量）：  【高频考点】【难点】方向相同或相反的非零向量称为平行向量。规定：零向量与任一向量平行。  教师强调：平行向量就是共线向量，这是因为任何一组平行向量都可以平移到同一条直线上。这并不意味着表示这些向量的有向线段必须在同一直线上，它们可以在不同直线上，只要方向相同或相反即可。  （2）相等向量：  【重要】长度相等且方向相同的向量称为相等向量。  记作：a=b\boldsymbol{a}=\boldsymbol{b}a=b。  教师通过几何画板展示：一组有向线段，它们长度相同，方向相同，但起点不同。引导学生判断它们是否相等。  学生得出结论：相等向量通过平移后，可以完全重合。向量的相等具有传递性。  （3）相反向量：  【基础】与向量a\boldsymbol{a}a长度相等，方向相反的向量，称为a\boldsymbol{a}a的相反向量。  记作：−a\boldsymbol{a}−a。  显然，AB→=−BA→\overrightarrow{AB}=\overrightarrow{BA}AB<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">=−BA<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">。  教师在黑板或PPT上构建一个概念关系图，帮助学生形成结构化的知识网络。  （五）例题精讲，巩固深化（约12分钟）  【非常重要】本环节旨在通过典型例题的剖析，帮助学生巩固新知，特别是突破平行向量与相等向量的难点。  例1：判断下列命题的真假，并说明理由。  （1）若两个向量相等，则它们的起点相同，终点也相同。  （2）若∣a∣=∣b∣|\boldsymbol{a}|=|\boldsymbol{b}|∣a∣=∣b∣，则a=b\boldsymbol{a}=\boldsymbol{b}a=b。  （3）若AB→=DC→\overrightarrow{AB}=\overrightarrow{DC}AB<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">=DC<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">，则四边形ABCD是平行四边形。  （4）若a∥b\boldsymbol{a}\parallel\boldsymbol{b}a∥b，b∥c\boldsymbol{b}\parallel\boldsymbol{c}b∥c，则a∥c\boldsymbol{a}\parallel\boldsymbol{c}a∥c。  （5）共线的向量，若起点不同，则终点一定不同。  解析：  （1）假命题。相等向量只要求大小相等、方向相同，与起点位置无关。例如，平行四边形ABCD中，AB→=DC→\overrightarrow{AB}=\overrightarrow{DC}AB<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">=DC<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">，但起点分别为A和D，终点分别为B和C。  （2）假命题。大小相等是向量相等的必要条件，但不是充分条件，还必须方向相同。  （3）假命题。当A、B、C、D四点共线时，AB→=DC→\overrightarrow{AB}=\overrightarrow{DC}AB<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">=DC<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">可以成立，但此时不能构成四边形，更不用说平行四边形了。正确的结论应该是：若AB→=DC→\overrightarrow{AB}=\overrightarrow{DC}AB<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">=DC<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">，则四边形ABCD是平行四边形或A、B、C、D四点共线。严谨的说法是：若AB→=DC→\overrightarrow{AB}=\overrightarrow{DC}AB<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">=DC<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">，则∣AB→∣=∣DC→∣|\overrightarrow{AB}|=|\overrightarrow{DC}|∣AB<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">∣=∣DC<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">∣且AB∥DC，即四边形ABCD是平行四边形。  （4）假命题。这是一个易错点。若b=0\boldsymbol{b}=\boldsymbol{0}b=0，则零向量与任何向量平行，但a\boldsymbol{a}a与c\boldsymbol{c}c不一定平行。例如，a\boldsymbol{a}a方向向东，c\boldsymbol{c}c方向向北，它们都与零向量平行，但a\boldsymbol{a}a与c\boldsymbol{c}c不平行。若加上条件b≠0\boldsymbol{b}\neq\boldsymbol{0}b=0，则命题为真。  （5）假命题。共线的向量，如果方向相同且长度也相同，那么它们就是相等向量，可以通过平移使得起点不同而终点相同。例如，在直线l上取两点A和B，再取两点C和D，使得AB→=CD→\overrightarrow{AB}=\overrightarrow{CD}AB<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">=CD<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">，则它们的终点分别是B和D，但B和D是不同的点。  例2：如图，O是正六边形ABCDEF的中心。分别写出与OA→\overrightarrow{OA}OA<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">、OB→\overrightarrow{OB}OB<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">、AB→\overrightarrow{AB}AB<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">相等的向量。  （教师提前准备好看的正六边形图形）  解析：引导学生观察正六边形的几何性质（对边平行且相等，中心到各顶点距离相等，中心与顶点连线夹角为60°等）。  （1）与OA→\overrightarrow{OA}OA<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">相等的向量：方向必须与OA→\overrightarrow{OA}OA<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">相同（由O指向A），且长度相等。观察可知，DO→\overrightarrow{DO}DO<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">、EF→\overrightarrow{EF}EF<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">、CB→\overrightarrow{CB}CB<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">都与OA→\overrightarrow{OA}OA<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">相等。  （2）与OB→\overrightarrow{OB}OB<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">相等的向量：EO→\overrightarrow{EO}EO<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">、FA→\overrightarrow{FA}FA<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">、DC→\overrightarrow{DC}DC<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">。  （3）与AB→\overrightarrow{AB}AB<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">相等的向量：FO→\overrightarrow{FO}FO<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">、OC→\overrightarrow{OC}OC<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">、ED→\overrightarrow{ED}ED<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">。  通过此题，学生能直观感受相等向量在图形中的表现，进一步强化“方向相同、大小相等”的概念。  例3：一辆汽车从A点出发向西行驶了100公里到达B点，然后又改变方向向西偏北50°行驶了200公里到达C点，最后又改变方向向东行驶了100公里到达D点。  （1）作出向量AB→\overrightarrow{AB}AB<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">、BC→\overrightarrow{BC}BC<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">、CD→\overrightarrow{CD}CD<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">。  （2）求向量AD→\overrightarrow{AD}AD<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">的模和方向。  解析：  （1）引导学生根据描述，选取合适的比例尺（如1cm代表100km）在纸上准确作出向量。  （2）分析发现，AB→\overrightarrow{AB}AB<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">和CD→\overrightarrow{CD}CD<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">长度相等，方向相反，所以它们互为相反向量。因此，从A到D的最终位移相当于从A点先向西100km，再向西偏北50°200km，再向东100km。由于向西100km和向东100km相互抵消，最终位移就等于从A点出发向西偏北50°行驶200km到达的点，即D点的位置就是C点的位置？严谨推导：  AD→=AB→+BC→+CD→=(AB→+CD→)+BC→=0+BC→=BC→\overrightarrow{AD}=\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CD}=(\overrightarrow{AB}+\overrightarrow{CD})+\overrightarrow{BC}=\boldsymbol{0}+\overrightarrow{BC}=\overrightarrow{BC}AD<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">=AB<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">+BC<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">+CD<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">=(AB<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">+CD<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">)+BC<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..7211102210..783.367151.zm00v40hv40z">=0+BC<pathd="M0241v40hc47.335.3847811012816.73227.763..3.22.7.54.31.3.52.3.5307.36.71120.2.815.52.52.31.74.25.55.511.5213.35.727114114.744.73984..5s73.760..5c6295.7911s39911c45.315.38540..5s58.374..5c4.7148.327..36.73.210.85.512.52.31.77.52.515.52..721