平面向量内积的坐标运算与距离公式

平面向量内积的坐标运算与距离公式 德清乾元职高 朱见锋 【教材分析】:本课是在平面向量坐标运算、内积定义基础上学习的，主要知识是平面向量内积的坐标运算与平面内两点间的距离公式，是后面学习曲线方程的重要公式和推导依据，是进一步学习相关数学知识的重要基础。 【教学目标】 1. 掌握平面向量内积的坐标表示，会应用平面向量内积的知识解决平面内有关长度、两向量的夹角和垂直的问题( 2. 能够根据平面向量的坐标，判断两向量是否垂直，求两向量的夹角等。 3. 通过学习平面向量的坐标表示，使学生进一步了解数学知识的相同性，培养学生辩证思维能力(提高学生数学知识的应用能力。 【教学重点】:平面向量内积的坐标公式式，平面向量垂直的充要条件，平面内两点间距离公式的应用( 【教学难点】:平面向量内积的坐标公式的推导和应用。 【教学方法】 本节课采用问题启发式教学和讲练结合的教学方法( 【教学过程】 环节 教学内容 师生互动 设计意图 前面我们学习了平面向量的平面直角坐标及其运算下面一起来教师提出问题( 学生回忆解答(师 回忆下这些知识:1.在平面直角坐标系中，是基向量，他e,e12 生共同回忆旧知识( 为知识迁 们的坐标如何表示,任意向量的坐标如何表示,的坐aa,b,,a 移做准备( 标如何表示, 2.上节课我们学习了向量的内积，是怎么定义的呢, ? = ab 师:对平面向量的复 内积的研究不能仅仅停cosa,b = 习 留在几何角度，还要寻导 3( 有哪些重要性质, 求其坐标表示(引出探入 究问题( ae?= = ,ab , ; a| |= ab???? 4.满足哪些运算律 abba(1)交换律:? =? spacing of 100mm. C, line line line nearby. D, spare wire unified terminal strip Terminal (terminal strip at the top of the vertical, horizontal terminal strip from the harness to Terminal Pai end at the beginning), length to ensure as far as Terminal and set aside. After e, form a wire harness, cable, line. When the line should be in the core wire for better identification and marking on the cards will be prepared in advance, alternate core core content for the cable numbering. F, finish line, according to the corresponding terminal locations, wire one by one from the bunch out, pulling line, should be kept parallel to each other and more. Core dark as possible, determined in accordance with specific conditions, it is required that: neat, beautiful, shapely, pleasing to the eye. 7.6.2 wiring technology, out of the core wire according to Terminal position, cut off the excess, with wire stripping pliers or electrician knife stripped of insulation, so that connection. Stripping should not damage the copper core. B, core wire on the oxide and chip-insulated electricians knife scrape. C, core processing has been completed, thread round long nose pliers in a clockwise direction. Circle with the screw in the direction of spin in the same direction, circle should be appropriate, it is best to slightly larger than the screws, bent round the circle and the length of root and appropriate so set screw is screwed on tight. D, when on the crimp-type terminals when wiring, threads do not have to bend. Insulating part should be careful not to press it into a Terminal, and make the loop doesn't make sense and to prevent bad head crimping, fell down, causing an open circuit. After wiring, may be tried. E, crimping Terminal crimping, wire and Terminal contact is good, solid, not subject to mechanical strain, and ensure the insulation level. F, cables, such as when there is a bare copper ground wire, bare copper wire with a plastic pipe should be set. G, when wiring wiring must be in radians. H, terminal screws (2)结合律:(λ)?,λ(?),?(λ); ababab (3)分配律:(，)?,?，? abcacbc 5.那么如何用坐标来表示?呢, ab 学生讨论并回答，问题为复 已知， 是直角坐标平面上的基向量，如果,(a，a)， aee1212 教师再提出的下列问习向量的线性 ,(b，b)，你能推导出? 的坐标公式吗, bab12题: 运算和向量的 内积而设计(通 探究过程 ，(1)(ae11 过学生的探究 ?,(a，a)?(b，b) abeeee12121212a)?(b，b))eee212212给出结论，比直新 ,ab?，ab?，ab?，ab?，eeeeeeee11122122 接给出更符合11121222课 是怎样进行运算的, 学生的特点，容知 又因为 (2)?，ee11易被学生接识 ?,1，?,1，?,0， eeeeee112212? ，?的eeee受(通过结论的2212讲 所以 探究，让学生初解 内积是怎样计算的, 步感受到无论?,ab，ab( ab1122 教师给出向量内积 是向量的线性 的直角坐标运算公定理 在直角坐标平面xoy中，如果,(a，a)，,(b，b)则 ab1212运算还是向量 式(并引导学生用文字 的内积运算，最 ?,ab，ab( ab1122叙述( 终都归结为直 即:两个向量的内积等于它们对应坐标的乘积的和( 角坐标运算( 因此可以推出两向量垂直的充要条件为 从而归纳总结 ?, a b，a b,0; ab1122出公式及数学 规律 在教师的引导下学问题:(1)若已知a=(a，a) ，你能用上面的定理求出| a | 吗, 12 生讨论得出( 解 因为 教师提出问题，稍2|aaa |=?=(a，a)?(a，a) 1212 加点拨( 22 ,a，a， 12学生讨论解答( 22教师总结得出这就a所以| |=a，a( 12 是根据向量的坐标求向这就是根据向量的坐标求向量长度的计算公式( 量长度的计算公式( 因此可推出两非零向量夹角余弦值公式为 教师例题分析讲解 ab，ab1122abcos‹，›,( 2222 a，ab，b1212学生边学边用 spacing of 100mm. C, line line line nearby. D, spare wire unified terminal strip Terminal (terminal strip at the top of the vertical, horizontal terminal strip from the harness to Terminal Pai end at the beginning), length to ensure as far as Terminal and set aside. After e, form a wire harness, cable, line. When the line should be in the core wire for better identification and marking on the cards will be prepared in advance, alternate core core content for the cable numbering. F, finish line, according to the corresponding terminal locations, wire one by one from the bunch out, pulling line, should be kept parallel to each other and more. Core dark as possible, determined in accordance with specific conditions, it is required that: neat, beautiful, shapely, pleasing to the eye. 7.6.2 wiring technology, out of the core wire according to Terminal position, cut off the excess, with wire stripping pliers or electrician knife stripped of insulation, so that connection. Stripping should not damage the copper core. B, core wire on the oxide and chip-insulated electricians knife scrape. C, core processing has been completed, thread round long nose pliers in a clockwise direction. Circle with the screw in the direction of spin in the same direction, circle should be appropriate, it is best to slightly larger than the screws, bent round the circle and the length of root and appropriate so set screw is screwed on tight. D, when on the crimp-type terminals when wiring, threads do not have to bend. Insulating part should be careful not to press it into a Terminal, and make the loop doesn't make sense and to prevent bad head crimping, fell down, causing an open circuit. After wiring, may be tried. E, crimping Terminal crimping, wire and Terminal contact is good, solid, not subject to mechanical strain, and ensure the insulation level. F, cables, such as when there is a bare copper ground wire, bare copper wire with a plastic pipe should be set. G, when wiring wiring must be in radians. H, terminal screws 例 通过例1例1 设,(3，,1)，,(1，,2)，求: ab 题 可让学生加深 (1) ?; (2) | |; aba讲 对向量内积的解 直角坐标运算(3) | |; (4)‹ ，›( bab 公式及向量的 解 (1) ?,3×1，(,1)×(,2),3，2,5; ab 长度公式的理 解和记忆( 22(2) | |,3，(,1),10; a 22 (3) ||,1，(—2),5; b (4) 因为 a,b52cos‹，›,,,， ab 2 10×5|a||b| π因为0?‹，›?所以‹，›,( abab, 4 配套学生练习: 练 学生练习巩固所学使刚刚学3已知,(0，2)，,(-2，2)，求: ab 习 知识 过的知识及时巩 得到应用(让学(1) ?; (2) | |; aba 固 生在边学边用 (3) | |; (4)‹ ，›( bab 中巩固知识，形 问题 教师提出问题( 成技能( ? (2)若已知A(x，y)，B(x，y)，如何求|AB| , 1122 学生讨论解答( 解 因为A(x，y)，B(x，y)，所以 1122 教师总结得出这就采用问题?AB,(x,x，y,y)( 2121 是根据两点的坐标求两诱导式让学生 ?点之间的距离公式( 更易理解和接22所以|AB|,(x,x)，(y,y)， 2121 受 这就是根据两点的坐标求两点之间的距离公式( 学生尝试解答(教 ?例2 已知A(2，,4)，B(,2，3)，求|AB|( 师针对学生的回答进行通过例2例 点评( 可让学生加深解 因为A(2，,4)，B(,2，3)，所以 题 对平面内两点?AB,(,2，3) ,(2，,4) 讲 间距离公式的 ,(,4，7)， 解 理解和记忆( ?22 所以| AB|,7，(,4),65( spacing of 100mm. C, line line line nearby. D, spare wire unified terminal strip Terminal (terminal strip at the top of the vertical, horizontal terminal strip from the harness to Terminal Pai end at the beginning), length to ensure as far as Terminal and set aside. After e, form a wire harness, cable, line. When the line should be in the core wire for better identification and marking on the cards will be prepared in advance, alternate core core content for the cable numbering. F, finish line, according to the corresponding terminal locations, wire one by one from the bunch out, pulling line, should be kept parallel to each other and more. Core dark as possible, determined in accordance with specific conditions, it is required that: neat, beautiful, shapely, pleasing to the eye. 7.6.2 wiring technology, out of the core wire according to Terminal position, cut off the excess, with wire stripping pliers or electrician knife stripped of insulation, so that connection. Stripping should not damage the copper core. B, core wire on the oxide and chip-insulated electricians knife scrape. C, core processing has been completed, thread round long nose pliers in a clockwise direction. Circle with the screw in the direction of spin in the same direction, circle should be appropriate, it is best to slightly larger than the screws, bent round the circle and the length of root and appropriate so set screw is screwed on tight. D, when on the crimp-type terminals when wiring, threads do not have to bend. Insulating part should be careful not to press it into a Terminal, and make the loop doesn't make sense and to prevent bad head crimping, fell down, causing an open circuit. After wiring, may be tried. E, crimping Terminal crimping, wire and Terminal contact is good, solid, not subject to mechanical strain, and ensure the insulation level. F, cables, such as when there is a bare copper ground wire, bare copper wire with a plastic pipe should be set. G, when wiring wiring must be in radians. H, terminal screws 练 学生练习: 已知A(2，1)，B(6，3)，C(5，0)，求:?ABC三边的长，学生练习，巩固所学习新知习 并判别?ABC是否为等腰三角形( 学知识 后紧跟练习，有巩 利于帮助学生 ??固 更好的梳理和例3已知A(1，2)，B(2，3)，C(,2，5)，求证:AB,AC( 总结本节所学 证明 因为 内容(有利于教?AB,(2,1，3,2),(1，1)， 师检验学生的 掌握情况( ?AC,(,2,1，5,2),(,3，3)， 可得 ??AB?AC,(1，1)?(,3，3),0( 教师点拨，学生解 答( ??所以AB , AC( 教师针对学生的回 答进行点评( 练习 快速判别上面的练习中的?ABC是否为等腰直角三角形, 新 课 本节课我们主要学习了平面向量内积的坐标运算与距离公式，常学生阅读课本,畅梳理总结 见的题型主要有: 谈本节课的收获，老师也可针对学生小 (1)直接用两向量的坐标计算平面向量的内积; 引导梳理，总结本节课薄弱或易错处结 (2)根据向量的坐标求该向量的模(长度); 主要的知识点( 进行强调和总 (3)根据两点坐标求这两点间的距离; 结( (4)运用平面向量的性质判定平面内两向量是否垂直, 作 教材 P56 练习A 组第 1 题; 巩固拓展( 业 教材 P57 练习 B 组第1题(选做)( spacing of 100mm. C, line line line nearby. D, spare wire unified terminal strip Terminal (terminal strip at the top of the vertical, horizontal terminal strip from the harness to Terminal Pai end at the beginning), length to ensure as far as Terminal and set aside. After e, form a wire harness, cable, line. When the line should be in the core wire for better identification and marking on the cards will be prepared in advance, alternate core core content for the cable numbering. F, finish line, according to the corresponding terminal locations, wire one by one from the bunch out, pulling line, should be kept parallel to each other and more. Core dark as possible, determined in accordance with specific conditions, it is required that: neat, beautiful, shapely, pleasing to the eye. 7.6.2 wiring technology, out of the core wire according to Terminal position, cut off the excess, with wire stripping pliers or electrician knife stripped of insulation, so that connection. Stripping should not damage the copper core. B, core wire on the oxide and chip-insulated electricians knife scrape. C, core processing has been completed, thread round long nose pliers in a clockwise direction. Circle with the screw in the direction of spin in the same direction, circle should be appropriate, it is best to slightly larger than the screws, bent round the circle and the length of root and appropriate so set screw is screwed on tight. D, when on the crimp-type terminals when wiring, threads do not have to bend. Insulating part should be careful not to press it into a Terminal, and make the loop doesn't make sense and to prevent bad head crimping, fell down, causing an open circuit. After wiring, may be tried. E, crimping Terminal crimping, wire and Terminal contact is good, solid, not subject to mechanical strain, and ensure the insulation level. F, cables, such as when there is a bare copper ground wire, bare copper wire with a plastic pipe should be set. G, when wiring wiring must be in radians. H, terminal screws 【设计理念】 数学学习是一个知识理解、迁移、转化的过程，因此要实现教学的有效性，必须知识点的迁移、转化， 引导学生充分利用自己已有的知识与经验，通过对问题的探究与解决，实现数学知识的转化，从而实现数 学知识的归纳和应用，达成教学目标。 spacing of 100mm. C, line line line nearby. D, spare wire unified terminal strip Terminal (terminal strip at the top of the vertical, horizontal terminal strip from the harness to Terminal Pai end at the beginning), length to ensure as far as Terminal and set aside. After e, form a wire harness, cable, line. When the line should be in the core wire for better identification and marking on the cards will be prepared in advance, alternate core core content for the cable numbering. F, finish line, according to the corresponding terminal locations, wire one by one from the bunch out, pulling line, should be kept parallel to each other and more. Core dark as possible, determined in accordance with specific conditions, it is required that: neat, beautiful, shapely, pleasing to the eye. 7.6.2 wiring technology, out of the core wire according to Terminal position, cut off the excess, with wire stripping pliers or electrician knife stripped of insulation, so that connection. Stripping should not damage the copper core. B, core wire on the oxide and chip-insulated electricians knife scrape. C, core processing has been completed, thread round long nose pliers in a clockwise direction. Circle with the screw in the direction of spin in the same direction, circle should be appropriate, it is best to slightly larger than the screws, bent round the circle and the length of root and appropriate so set screw is screwed on tight. D, when on the crimp-type terminals when wiring, threads do not have to bend. Insulating part should be careful not to press it into a Terminal, and make the loop doesn't make sense and to prevent bad head crimping, fell down, causing an open circuit. After wiring, may be tried. E, crimping Terminal crimping, wire and Terminal contact is good, solid, not subject to mechanical strain, and ensure the insulation level. F, cables, such as when there is a bare copper ground wire, bare copper wire with a plastic pipe should be set. G, when wiring wiring must be in radians. H, terminal screws