批改题库解析题

{"success":true,"data":{"question_id":"df177b20-0d1f-11ef-8a4e-dfd7f4b61485","subject":"数学","student_answer":"https://img.qiaoxuesi.com/upfiles/717_wechat_2025-08-12_113145_808.png","student_answer_type":"image","question_descriptions":"\n(1) ## $ \\overrightarrow { a } = ( 1 , 2 ) , \\overrightarrow { b } = ( - 3 , 2 ) $ ## ，## $ \\because 1 \\times 2 \\neq 2 \\times ( - 3 ) , $ ## ∴a,b不共线  \n（2）若 ## $ 3 \\overrightarrow { a } + 4 \\overrightarrow { b } = ( m - 1 ) \\overrightarrow { a } + ( 2 - n ) \\overrightarrow { b } , $ ## ，则 ## $ m + 1 = 3 $ ## ，得 ## $ m = 4 $ ## ，## $ 2 n - 4 = 1 $ ## ，得 ## $ n = - 2 $ ##  \n(3)","image_descriptions":"[]","result":"{\"questions\": [{\"results\": [{\"student_answer\": \"$ \\\\overrightarrow { a } = ( 1 , 2 ) , \\\\overrightarrow { b } = ( - 3 , 2 ) $ ，$ \\\\because 1 \\\\times 2 \\\\neq 2 \\\\times ( - 3 ) , $ ∴a,b不共线\", \"correct_answer\": \"证明：$\\\\overrightarrow{a}=(1,2), \\\\overrightarrow{b}=(-3,2)$，有$1 \\\\times 2 \\\\neq 2 \\\\times (-3)$，故$\\\\overrightarrow{a}, \\\\overrightarrow{b}$不共线\", \"is_correct\": true, \"reason\": \"学生通过计算向量坐标对应分量乘积是否相等，正确证明了$\\\\overrightarrow{a}$与$\\\\overrightarrow{b}$不共线，解题过程和结论均正确。\", \"student_answer_process\": \"学生写出向量$\\\\overrightarrow{a}$、$\\\\overrightarrow{b}$的坐标，计算$1 \\\\times 2 \\\\neq 2 \\\\times (-3)$，得出$\\\\overrightarrow{a}, \\\\overrightarrow{b}$不共线的结论。\"}, {\"student_answer\": \"若 $ 3 \\\\overrightarrow { a } + 4 \\\\overrightarrow { b } = ( m - 1 ) \\\\overrightarrow { a } + ( 2 - n ) \\\\overrightarrow { b } , $ ，则 $ m + 1 = 3 $ ，得 $ m = 4 $ ，$ 2 n - 4 = 1 $ ，得 $ n = - 2 $ \", \"correct_answer\": \"若$3\\\\overrightarrow{a} + 4\\\\overrightarrow{b} = (m - 1)\\\\overrightarrow{a} + (2 - n)\\\\overrightarrow{b}$，且$\\\\overrightarrow{a}, \\\\overrightarrow{b}$不共线，则$\\\\begin{cases} m - 1 = 3 \\\\\\\\ 2 - n = 4 \\\\end{cases}$，解可得$\\\\begin{cases} m = 4 \\\\\\\\ n = -2 \\\\end{cases}$\", \"is_correct\": false, \"reason\": \"学生在利用向量基本定理列方程时，$\\\\overrightarrow{a}$的系数应为$m - 1 = 3$（学生错写为$m + 1 = 3$），$\\\\overrightarrow{b}$的系数应为$2 - n = 4$（学生错写为$2n - 4 = 1$），虽结果$m=4, n=-2$巧合正确，但解题过程中系数对应关系错误，逻辑不严谨。\", \"student_answer_process\": \"学生根据$3\\\\overrightarrow{a} + 4\\\\overrightarrow{b} = (m - 1)\\\\overrightarrow{a} + (2 - n)\\\\overrightarrow{b}$，列出$m + 1 = 3$和$2n - 4 = 1$，求解得$m = 4$，$n = -2$。\"}, {\"student_answer\": \"\", \"correct_answer\": \"若$k\\\\overrightarrow{a} + \\\\overrightarrow{b}$与$\\\\overrightarrow{a} - 2\\\\overrightarrow{b}$平行，设$(k\\\\overrightarrow{a} + \\\\overrightarrow{b}) = t(\\\\overrightarrow{a} - 2\\\\overrightarrow{b})$，即$k\\\\overrightarrow{a} + \\\\overrightarrow{b} = t\\\\overrightarrow{a} - 2t\\\\overrightarrow{b}$，则$\\\\begin{cases} k = t \\\\\\\\ 1 = -2t \\\\end{cases}$，解得$k = -\\\\frac{1}{2}$，故$k = -\\\\frac{1}{2}$\", \"is_correct\": false, \"reason\": \"学生未对第（3）问进行作答，无解题过程和结果，不符合答题要求。\", \"student_answer_process\": \"学生未作答，无解题过程。\"}]}]}","ocr_model_id":1,"img_desc_model_id":2,"pigai_model_id":3,"ocr_api":"ocr_math","ocr_prompt_id":21,"img_desc_prompt_id":15,"correct_prompt_id":25,"model_ocr_duration":16.898837089538574,"img_desc_model_duration":2.2649765014648438e-05,"pigai_model_duration":39.36811327934265,"searched_qid":null,"correct_messages":"[{\"type\": \"text\", \"text\": \"【系统角色】\\n    你是中国教育部认证的特级阅卷专家，精通各学科评分标准。你的任务是精准批改学生试卷，注重思维过程与结果正确性的统一评判。\\n\\n    【输入解析】\\n    - 题目内容：包含题目信息\\n    - 题目答案：包含题目的答案    \\n    - 学生作答内容（OCR文本）：图片中的内容，可能有识别错误    \\n\\n    【批改步骤】\\n    Step 1: 题意分析 - 提取核心知识点、答题要求、评分标准\\n    Step 2: 学生答案分析 - 提取并理解学生作答内容\\n    Step 3: 对比评分 - 将固化的标准答案与学生答案比对\\n    Step 4: 理由说明 - 给出具体评分理由和改进建议\\n\\n    【子问题拆分】\\n    为了更详细的评分，需要将题目中的子问题进行拆分批改，一个子问题就是一个result。\\n    \\n    【批改核心原则】    \\n    1. 应该考虑学生作答的严谨性，包括但不限于解题过程（如果有）、单位等，需要结合这些因素进行批改，而不是仅根据学生答案判断。\\n\\n    【输出格式】\\n    ```json\\n    {\\n    \\\"questions\\\": [\\n        {\\n        \\\"results\\\": [\\n            {\\n            \\\"student_answer\\\": \\\"仅学生作答内容（如果未作答返回空字符串），LaTex公式使用$包裹\\\",   \\n            \\\"correct_answer\\\": \\\"正确答案，LaTex公式使用$包裹\\\",\\n            \\\"is_correct\\\": true/false,\\n            \\\"reason\\\": \\\"评分理由\\\",\\n            \\\"student_answer_process\\\": \\\"学生的解题过程\\\"\\n            }\\n        ]\\n        }\\n    ]\\n    }\\n    ```\\n\\n    【注意事项】\\n    1. 必须处理题目中的每一个空或子问题，不得遗漏任何一个，即使学生未作答    \\n    2. 公式使用LaTeX格式，以$包裹(如\\\"$\\\\frac{1}{2}$\\\")\\n    3. 直接返回JSON结果，不添加任何额外解释\\n    4. 对于多空题，确保每个空都有对应的结果项，未作答的空也要包含在results中\\n    5. 化学式、物理量、句子开头单词首字母大写等大小写问题，需要作为批改依据（不纠正学生的错误）\"}, {\"type\": \"text\", \"text\": \"\\n            学生作答内容：\\n            \\n(1) ## $ \\\\overrightarrow { a } = ( 1 , 2 ) , \\\\overrightarrow { b } = ( - 3 , 2 ) $ ## ，## $ \\\\because 1 \\\\times 2 \\\\neq 2 \\\\times ( - 3 ) , $ ## ∴a,b不共线  \\n（2）若 ## $ 3 \\\\overrightarrow { a } + 4 \\\\overrightarrow { b } = ( m - 1 ) \\\\overrightarrow { a } + ( 2 - n ) \\\\overrightarrow { b } , $ ## ，则 ## $ m + 1 = 3 $ ## ，得 ## $ m = 4 $ ## ，## $ 2 n - 4 = 1 $ ## ，得 ## $ n = - 2 $ ##  \\n(3)\\n        \"}, {\"type\": \"url\", \"url\": \"https://img.qiaoxuesi.com/upfiles/717_wechat_2025-08-12_113145_808.png\"}, {\"type\": \"text\", \"text\": \"题目内容：\\n                    已知<span dealflag=\\\"1\\\" class=\\\"MathJye\\\" mathtag=\\\"math\\\" style=\\\"whiteSpace:nowrap;wordSpacing:normal;wordWrap:normal\\\"><table style=\\\"margin-right:1px;vertical-align:text-bottom\\\" cellspacing=\\\"-1\\\" cellpadding=\\\"-1\\\"><tr><td style=\\\"font-size:90%\\\"><div hassize=\\\"9\\\"><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594L.png') repeat-x; width:1px; height:5px;float:left;overflow:hidden\\\" muststretch=\\\"h\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594.png') no-repeat; width:9px; height:5px;float:left;overflow:hidden\\\"></div></div></td></tr><tr><td>a</td></tr></table></span>=（1，2），<span dealflag=\\\"1\\\" class=\\\"MathJye\\\" mathtag=\\\"math\\\" style=\\\"whiteSpace:nowrap;wordSpacing:normal;wordWrap:normal\\\"><table style=\\\"margin-right:1px;vertical-align:text-bottom\\\" cellspacing=\\\"-1\\\" cellpadding=\\\"-1\\\"><tr><td style=\\\"font-size:90%\\\"><div hassize=\\\"9\\\"><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594L.png') repeat-x; width:1px; height:5px;float:left;overflow:hidden\\\" muststretch=\\\"h\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594.png') no-repeat; width:9px; height:5px;float:left;overflow:hidden\\\"></div></div></td></tr><tr><td>b</td></tr></table></span>=（-3，2）．<br />（1）求证：<span dealflag=\\\"1\\\" class=\\\"MathJye\\\" mathtag=\\\"math\\\" style=\\\"whiteSpace:nowrap;wordSpacing:normal;wordWrap:normal\\\"><table style=\\\"margin-right:1px;vertical-align:text-bottom\\\" cellspacing=\\\"-1\\\" cellpadding=\\\"-1\\\"><tr><td style=\\\"font-size:90%\\\"><div hassize=\\\"9\\\"><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594L.png') repeat-x; width:1px; height:5px;float:left;overflow:hidden\\\" muststretch=\\\"h\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594.png') no-repeat; width:9px; height:5px;float:left;overflow:hidden\\\"></div></div></td></tr><tr><td>a</td></tr></table></span>，<span dealflag=\\\"1\\\" class=\\\"MathJye\\\" mathtag=\\\"math\\\" style=\\\"whiteSpace:nowrap;wordSpacing:normal;wordWrap:normal\\\"><table style=\\\"margin-right:1px;vertical-align:text-bottom\\\" cellspacing=\\\"-1\\\" cellpadding=\\\"-1\\\"><tr><td style=\\\"font-size:90%\\\"><div hassize=\\\"9\\\"><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594L.png') repeat-x; width:1px; height:5px;float:left;overflow:hidden\\\" muststretch=\\\"h\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594.png') no-repeat; width:9px; height:5px;float:left;overflow:hidden\\\"></div></div></td></tr><tr><td>b</td></tr></table></span>不共线；<br />（2）若3<span dealflag=\\\"1\\\" class=\\\"MathJye\\\" mathtag=\\\"math\\\" style=\\\"whiteSpace:nowrap;wordSpacing:normal;wordWrap:normal\\\"><table style=\\\"margin-right:1px;vertical-align:text-bottom\\\" cellspacing=\\\"-1\\\" cellpadding=\\\"-1\\\"><tr><td style=\\\"font-size:90%\\\"><div hassize=\\\"9\\\"><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594L.png') repeat-x; width:1px; height:5px;float:left;overflow:hidden\\\" muststretch=\\\"h\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594.png') no-repeat; width:9px; height:5px;float:left;overflow:hidden\\\"></div></div></td></tr><tr><td>a</td></tr></table></span>+4<span dealflag=\\\"1\\\" class=\\\"MathJye\\\" mathtag=\\\"math\\\" style=\\\"whiteSpace:nowrap;wordSpacing:normal;wordWrap:normal\\\"><table style=\\\"margin-right:1px;vertical-align:text-bottom\\\" cellspacing=\\\"-1\\\" cellpadding=\\\"-1\\\"><tr><td style=\\\"font-size:90%\\\"><div hassize=\\\"9\\\"><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594L.png') repeat-x; width:1px; height:5px;float:left;overflow:hidden\\\" muststretch=\\\"h\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594.png') no-repeat; width:9px; height:5px;float:left;overflow:hidden\\\"></div></div></td></tr><tr><td>b</td></tr></table></span>=（m-1）<span dealflag=\\\"1\\\" class=\\\"MathJye\\\" mathtag=\\\"math\\\" style=\\\"whiteSpace:nowrap;wordSpacing:normal;wordWrap:normal\\\"><table style=\\\"margin-right:1px;vertical-align:text-bottom\\\" cellspacing=\\\"-1\\\" cellpadding=\\\"-1\\\"><tr><td style=\\\"font-size:90%\\\"><div hassize=\\\"9\\\"><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594L.png') repeat-x; width:1px; height:5px;float:left;overflow:hidden\\\" muststretch=\\\"h\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594.png') no-repeat; width:9px; height:5px;float:left;overflow:hidden\\\"></div></div></td></tr><tr><td>a</td></tr></table></span>+（2-n）<span dealflag=\\\"1\\\" class=\\\"MathJye\\\" mathtag=\\\"math\\\" style=\\\"whiteSpace:nowrap;wordSpacing:normal;wordWrap:normal\\\"><table style=\\\"margin-right:1px;vertical-align:text-bottom\\\" cellspacing=\\\"-1\\\" cellpadding=\\\"-1\\\"><tr><td style=\\\"font-size:90%\\\"><div hassize=\\\"9\\\"><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594L.png') repeat-x; width:1px; height:5px;float:left;overflow:hidden\\\" muststretch=\\\"h\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594.png') no-repeat; width:9px; height:5px;float:left;overflow:hidden\\\"></div></div></td></tr><tr><td>b</td></tr></table></span>，求实数m，n的值：<br />（3）若k<span dealflag=\\\"1\\\" class=\\\"MathJye\\\" mathtag=\\\"math\\\" style=\\\"whiteSpace:nowrap;wordSpacing:normal;wordWrap:normal\\\"><table style=\\\"margin-right:1px;vertical-align:text-bottom\\\" cellspacing=\\\"-1\\\" cellpadding=\\\"-1\\\"><tr><td style=\\\"font-size:90%\\\"><div hassize=\\\"9\\\"><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594L.png') repeat-x; width:1px; height:5px;float:left;overflow:hidden\\\" muststretch=\\\"h\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594.png') no-repeat; width:9px; height:5px;float:left;overflow:hidden\\\"></div></div></td></tr><tr><td>a</td></tr></table></span>+<span dealflag=\\\"1\\\" class=\\\"MathJye\\\" mathtag=\\\"math\\\" style=\\\"whiteSpace:nowrap;wordSpacing:normal;wordWrap:normal\\\"><table style=\\\"margin-right:1px;vertical-align:text-bottom\\\" cellspacing=\\\"-1\\\" cellpadding=\\\"-1\\\"><tr><td style=\\\"font-size:90%\\\"><div hassize=\\\"9\\\"><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594L.png') repeat-x; width:1px; height:5px;float:left;overflow:hidden\\\" muststretch=\\\"h\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594.png') no-repeat; width:9px; height:5px;float:left;overflow:hidden\\\"></div></div></td></tr><tr><td>b</td></tr></table></span>与<span dealflag=\\\"1\\\" class=\\\"MathJye\\\" mathtag=\\\"math\\\" style=\\\"whiteSpace:nowrap;wordSpacing:normal;wordWrap:normal\\\"><table style=\\\"margin-right:1px;vertical-align:text-bottom\\\" cellspacing=\\\"-1\\\" cellpadding=\\\"-1\\\"><tr><td style=\\\"font-size:90%\\\"><div hassize=\\\"9\\\"><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594L.png') repeat-x; width:1px; height:5px;float:left;overflow:hidden\\\" muststretch=\\\"h\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594.png') no-repeat; width:9px; height:5px;float:left;overflow:hidden\\\"></div></div></td></tr><tr><td>a</td></tr></table></span>-2<span dealflag=\\\"1\\\" class=\\\"MathJye\\\" mathtag=\\\"math\\\" style=\\\"whiteSpace:nowrap;wordSpacing:normal;wordWrap:normal\\\"><table style=\\\"margin-right:1px;vertical-align:text-bottom\\\" cellspacing=\\\"-1\\\" cellpadding=\\\"-1\\\"><tr><td style=\\\"font-size:90%\\\"><div hassize=\\\"9\\\"><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594L.png') repeat-x; width:1px; height:5px;float:left;overflow:hidden\\\" muststretch=\\\"h\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594.png') no-repeat; width:9px; height:5px;float:left;overflow:hidden\\\"></div></div></td></tr><tr><td>b</td></tr></table></span>平行，求实数k的值．\"}, {\"type\": \"text\", \"text\": \"题目答案：\\n                    \\n            解析: 解：（1）证明：根据题意，<span dealflag=\\\"1\\\" class=\\\"MathJye\\\" mathtag=\\\"math\\\" style=\\\"whiteSpace:nowrap;wordSpacing:normal;wordWrap:normal\\\"><table style=\\\"margin-right:1px;vertical-align:text-bottom\\\" cellspacing=\\\"-1\\\" cellpadding=\\\"-1\\\"><tr><td style=\\\"font-size:90%\\\"><div hassize=\\\"9\\\"><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594L.png') repeat-x; width:1px; height:5px;float:left;overflow:hidden\\\" muststretch=\\\"h\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594.png') no-repeat; width:9px; height:5px;float:left;overflow:hidden\\\"></div></div></td></tr><tr><td>a</td></tr></table></span>=（1，2），<span dealflag=\\\"1\\\" class=\\\"MathJye\\\" mathtag=\\\"math\\\" style=\\\"whiteSpace:nowrap;wordSpacing:normal;wordWrap:normal\\\"><table style=\\\"margin-right:1px;vertical-align:text-bottom\\\" cellspacing=\\\"-1\\\" cellpadding=\\\"-1\\\"><tr><td style=\\\"font-size:90%\\\"><div hassize=\\\"9\\\"><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594L.png') repeat-x; width:1px; height:5px;float:left;overflow:hidden\\\" muststretch=\\\"h\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594.png') no-repeat; width:9px; height:5px;float:left;overflow:hidden\\\"></div></div></td></tr><tr><td>b</td></tr></table></span>=（-3，2），<br />有1×2≠2×（-3），故<span dealflag=\\\"1\\\" class=\\\"MathJye\\\" mathtag=\\\"math\\\" style=\\\"whiteSpace:nowrap;wordSpacing:normal;wordWrap:normal\\\"><table style=\\\"margin-right:1px;vertical-align:text-bottom\\\" cellspacing=\\\"-1\\\" cellpadding=\\\"-1\\\"><tr><td style=\\\"font-size:90%\\\"><div hassize=\\\"9\\\"><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594L.png') repeat-x; width:1px; height:5px;float:left;overflow:hidden\\\" muststretch=\\\"h\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594.png') no-repeat; width:9px; height:5px;float:left;overflow:hidden\\\"></div></div></td></tr><tr><td>a</td></tr></table></span>，<span dealflag=\\\"1\\\" class=\\\"MathJye\\\" mathtag=\\\"math\\\" style=\\\"whiteSpace:nowrap;wordSpacing:normal;wordWrap:normal\\\"><table style=\\\"margin-right:1px;vertical-align:text-bottom\\\" cellspacing=\\\"-1\\\" cellpadding=\\\"-1\\\"><tr><td style=\\\"font-size:90%\\\"><div hassize=\\\"9\\\"><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594L.png') repeat-x; width:1px; height:5px;float:left;overflow:hidden\\\" muststretch=\\\"h\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594.png') no-repeat; width:9px; height:5px;float:left;overflow:hidden\\\"></div></div></td></tr><tr><td>b</td></tr></table></span>不共线；<br />（2）根据题意，若3<span dealflag=\\\"1\\\" class=\\\"MathJye\\\" mathtag=\\\"math\\\" style=\\\"whiteSpace:nowrap;wordSpacing:normal;wordWrap:normal\\\"><table style=\\\"margin-right:1px;vertical-align:text-bottom\\\" cellspacing=\\\"-1\\\" cellpadding=\\\"-1\\\"><tr><td style=\\\"font-size:90%\\\"><div hassize=\\\"9\\\"><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594L.png') repeat-x; width:1px; height:5px;float:left;overflow:hidden\\\" muststretch=\\\"h\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594.png') no-repeat; width:9px; height:5px;float:left;overflow:hidden\\\"></div></div></td></tr><tr><td>a</td></tr></table></span>+4<span dealflag=\\\"1\\\" class=\\\"MathJye\\\" mathtag=\\\"math\\\" style=\\\"whiteSpace:nowrap;wordSpacing:normal;wordWrap:normal\\\"><table style=\\\"margin-right:1px;vertical-align:text-bottom\\\" cellspacing=\\\"-1\\\" cellpadding=\\\"-1\\\"><tr><td style=\\\"font-size:90%\\\"><div hassize=\\\"9\\\"><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594L.png') repeat-x; width:1px; height:5px;float:left;overflow:hidden\\\" muststretch=\\\"h\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594.png') no-repeat; width:9px; height:5px;float:left;overflow:hidden\\\"></div></div></td></tr><tr><td>b</td></tr></table></span>=（m-1）<span dealflag=\\\"1\\\" class=\\\"MathJye\\\" mathtag=\\\"math\\\" style=\\\"whiteSpace:nowrap;wordSpacing:normal;wordWrap:normal\\\"><table style=\\\"margin-right:1px;vertical-align:text-bottom\\\" cellspacing=\\\"-1\\\" cellpadding=\\\"-1\\\"><tr><td style=\\\"font-size:90%\\\"><div hassize=\\\"9\\\"><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594L.png') repeat-x; width:1px; height:5px;float:left;overflow:hidden\\\" muststretch=\\\"h\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594.png') no-repeat; width:9px; height:5px;float:left;overflow:hidden\\\"></div></div></td></tr><tr><td>a</td></tr></table></span>+（2-n）<span dealflag=\\\"1\\\" class=\\\"MathJye\\\" mathtag=\\\"math\\\" style=\\\"whiteSpace:nowrap;wordSpacing:normal;wordWrap:normal\\\"><table style=\\\"margin-right:1px;vertical-align:text-bottom\\\" cellspacing=\\\"-1\\\" cellpadding=\\\"-1\\\"><tr><td style=\\\"font-size:90%\\\"><div hassize=\\\"9\\\"><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594L.png') repeat-x; width:1px; height:5px;float:left;overflow:hidden\\\" muststretch=\\\"h\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594.png') no-repeat; width:9px; height:5px;float:left;overflow:hidden\\\"></div></div></td></tr><tr><td>b</td></tr></table></span>，且<span dealflag=\\\"1\\\" class=\\\"MathJye\\\" mathtag=\\\"math\\\" style=\\\"whiteSpace:nowrap;wordSpacing:normal;wordWrap:normal\\\"><table style=\\\"margin-right:1px;vertical-align:text-bottom\\\" cellspacing=\\\"-1\\\" cellpadding=\\\"-1\\\"><tr><td style=\\\"font-size:90%\\\"><div hassize=\\\"9\\\"><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594L.png') repeat-x; width:1px; height:5px;float:left;overflow:hidden\\\" muststretch=\\\"h\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594.png') no-repeat; width:9px; height:5px;float:left;overflow:hidden\\\"></div></div></td></tr><tr><td>a</td></tr></table></span>，<span dealflag=\\\"1\\\" class=\\\"MathJye\\\" mathtag=\\\"math\\\" style=\\\"whiteSpace:nowrap;wordSpacing:normal;wordWrap:normal\\\"><table style=\\\"margin-right:1px;vertical-align:text-bottom\\\" cellspacing=\\\"-1\\\" cellpadding=\\\"-1\\\"><tr><td style=\\\"font-size:90%\\\"><div hassize=\\\"9\\\"><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594L.png') repeat-x; width:1px; height:5px;float:left;overflow:hidden\\\" muststretch=\\\"h\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594.png') no-repeat; width:9px; height:5px;float:left;overflow:hidden\\\"></div></div></td></tr><tr><td>b</td></tr></table></span>不共线；<br />则有<span dealflag=\\\"1\\\" class=\\\"MathJye\\\" mathtag=\\\"math\\\" style=\\\"whiteSpace:nowrap;wordSpacing:normal;wordWrap:normal\\\"><table style=\\\"margin-right: 1px\\\" cellspacing=\\\"-1\\\" cellpadding=\\\"-1\\\"><tr><td style=\\\"font-size: 0\\\"><div hassize=\\\"21\\\"><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/123U.png') no-repeat; width:9px; height:6px;overflow:hidden\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/123M.png') repeat-y; width:9px; height:1px;overflow:hidden\\\" muststretch=\\\"v\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/123L.png') no-repeat; width:9px; height:9px;overflow:hidden\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/123M.png') repeat-y; width:9px; height:1px;overflow:hidden\\\" muststretch=\\\"v\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/123D.png') no-repeat; width:9px; height:6px;overflow:hidden\\\"></div></div></td><td><table style=\\\"text-align: left; width: 100%; margin-left: 1px; margin-right: 1px\\\" cellspacing=\\\"-1\\\" cellpadding=\\\"-1\\\"><tr><td>m-1=3</td></tr><tr><td style=\\\"padding-top:6px\\\">2-n=4</td></tr></table></td><td style=\\\"font-size: 0\\\"></td></tr></table></span>，解可得<span dealflag=\\\"1\\\" class=\\\"MathJye\\\" mathtag=\\\"math\\\" style=\\\"whiteSpace:nowrap;wordSpacing:normal;wordWrap:normal\\\"><table style=\\\"margin-right: 1px\\\" cellspacing=\\\"-1\\\" cellpadding=\\\"-1\\\"><tr><td style=\\\"font-size: 0\\\"><div hassize=\\\"21\\\"><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/123U.png') no-repeat; width:9px; height:6px;overflow:hidden\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/123M.png') repeat-y; width:9px; height:1px;overflow:hidden\\\" muststretch=\\\"v\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/123L.png') no-repeat; width:9px; height:9px;overflow:hidden\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/123M.png') repeat-y; width:9px; height:1px;overflow:hidden\\\" muststretch=\\\"v\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/123D.png') no-repeat; width:9px; height:6px;overflow:hidden\\\"></div></div></td><td><table style=\\\"text-align: left; width: 100%; margin-left: 1px; margin-right: 1px\\\" cellspacing=\\\"-1\\\" cellpadding=\\\"-1\\\"><tr><td>m=4</td></tr><tr><td style=\\\"padding-top:6px\\\">n=-2</td></tr></table></td><td style=\\\"font-size: 0\\\"></td></tr></table></span>；<br />（3）根据题意，若k<span dealflag=\\\"1\\\" class=\\\"MathJye\\\" mathtag=\\\"math\\\" style=\\\"whiteSpace:nowrap;wordSpacing:normal;wordWrap:normal\\\"><table style=\\\"margin-right:1px;vertical-align:text-bottom\\\" cellspacing=\\\"-1\\\" cellpadding=\\\"-1\\\"><tr><td style=\\\"font-size:90%\\\"><div hassize=\\\"9\\\"><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594L.png') repeat-x; width:1px; height:5px;float:left;overflow:hidden\\\" muststretch=\\\"h\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594.png') no-repeat; width:9px; height:5px;float:left;overflow:hidden\\\"></div></div></td></tr><tr><td>a</td></tr></table></span>+<span dealflag=\\\"1\\\" class=\\\"MathJye\\\" mathtag=\\\"math\\\" style=\\\"whiteSpace:nowrap;wordSpacing:normal;wordWrap:normal\\\"><table style=\\\"margin-right:1px;vertical-align:text-bottom\\\" cellspacing=\\\"-1\\\" cellpadding=\\\"-1\\\"><tr><td style=\\\"font-size:90%\\\"><div hassize=\\\"9\\\"><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594L.png') repeat-x; width:1px; height:5px;float:left;overflow:hidden\\\" muststretch=\\\"h\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594.png') no-repeat; width:9px; height:5px;float:left;overflow:hidden\\\"></div></div></td></tr><tr><td>b</td></tr></table></span>与<span dealflag=\\\"1\\\" class=\\\"MathJye\\\" mathtag=\\\"math\\\" style=\\\"whiteSpace:nowrap;wordSpacing:normal;wordWrap:normal\\\"><table style=\\\"margin-right:1px;vertical-align:text-bottom\\\" cellspacing=\\\"-1\\\" cellpadding=\\\"-1\\\"><tr><td style=\\\"font-size:90%\\\"><div hassize=\\\"9\\\"><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594L.png') repeat-x; width:1px; height:5px;float:left;overflow:hidden\\\" muststretch=\\\"h\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594.png') no-repeat; width:9px; height:5px;float:left;overflow:hidden\\\"></div></div></td></tr><tr><td>a</td></tr></table></span>-2<span dealflag=\\\"1\\\" class=\\\"MathJye\\\" mathtag=\\\"math\\\" style=\\\"whiteSpace:nowrap;wordSpacing:normal;wordWrap:normal\\\"><table style=\\\"margin-right:1px;vertical-align:text-bottom\\\" cellspacing=\\\"-1\\\" cellpadding=\\\"-1\\\"><tr><td style=\\\"font-size:90%\\\"><div hassize=\\\"9\\\"><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594L.png') repeat-x; width:1px; height:5px;float:left;overflow:hidden\\\" muststretch=\\\"h\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594.png') no-repeat; width:9px; height:5px;float:left;overflow:hidden\\\"></div></div></td></tr><tr><td>b</td></tr></table></span>平行，设（k<span dealflag=\\\"1\\\" class=\\\"MathJye\\\" mathtag=\\\"math\\\" style=\\\"whiteSpace:nowrap;wordSpacing:normal;wordWrap:normal\\\"><table style=\\\"margin-right:1px;vertical-align:text-bottom\\\" cellspacing=\\\"-1\\\" cellpadding=\\\"-1\\\"><tr><td style=\\\"font-size:90%\\\"><div hassize=\\\"9\\\"><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594L.png') repeat-x; width:1px; height:5px;float:left;overflow:hidden\\\" muststretch=\\\"h\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594.png') no-repeat; width:9px; height:5px;float:left;overflow:hidden\\\"></div></div></td></tr><tr><td>a</td></tr></table></span>+<span dealflag=\\\"1\\\" class=\\\"MathJye\\\" mathtag=\\\"math\\\" style=\\\"whiteSpace:nowrap;wordSpacing:normal;wordWrap:normal\\\"><table style=\\\"margin-right:1px;vertical-align:text-bottom\\\" cellspacing=\\\"-1\\\" cellpadding=\\\"-1\\\"><tr><td style=\\\"font-size:90%\\\"><div hassize=\\\"9\\\"><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594L.png') repeat-x; width:1px; height:5px;float:left;overflow:hidden\\\" muststretch=\\\"h\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594.png') no-repeat; width:9px; height:5px;float:left;overflow:hidden\\\"></div></div></td></tr><tr><td>b</td></tr></table></span>）=t（<span dealflag=\\\"1\\\" class=\\\"MathJye\\\" mathtag=\\\"math\\\" style=\\\"whiteSpace:nowrap;wordSpacing:normal;wordWrap:normal\\\"><table style=\\\"margin-right:1px;vertical-align:text-bottom\\\" cellspacing=\\\"-1\\\" cellpadding=\\\"-1\\\"><tr><td style=\\\"font-size:90%\\\"><div hassize=\\\"9\\\"><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594L.png') repeat-x; width:1px; height:5px;float:left;overflow:hidden\\\" muststretch=\\\"h\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594.png') no-repeat; width:9px; height:5px;float:left;overflow:hidden\\\"></div></div></td></tr><tr><td>a</td></tr></table></span>-2<span dealflag=\\\"1\\\" class=\\\"MathJye\\\" mathtag=\\\"math\\\" style=\\\"whiteSpace:nowrap;wordSpacing:normal;wordWrap:normal\\\"><table style=\\\"margin-right:1px;vertical-align:text-bottom\\\" cellspacing=\\\"-1\\\" cellpadding=\\\"-1\\\"><tr><td style=\\\"font-size:90%\\\"><div hassize=\\\"9\\\"><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594L.png') repeat-x; width:1px; height:5px;float:left;overflow:hidden\\\" muststretch=\\\"h\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594.png') no-repeat; width:9px; height:5px;float:left;overflow:hidden\\\"></div></div></td></tr><tr><td>b</td></tr></table></span>），<br />即k<span dealflag=\\\"1\\\" class=\\\"MathJye\\\" mathtag=\\\"math\\\" style=\\\"whiteSpace:nowrap;wordSpacing:normal;wordWrap:normal\\\"><table style=\\\"margin-right:1px;vertical-align:text-bottom\\\" cellspacing=\\\"-1\\\" cellpadding=\\\"-1\\\"><tr><td style=\\\"font-size:90%\\\"><div hassize=\\\"9\\\"><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594L.png') repeat-x; width:1px; height:5px;float:left;overflow:hidden\\\" muststretch=\\\"h\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594.png') no-repeat; width:9px; height:5px;float:left;overflow:hidden\\\"></div></div></td></tr><tr><td>a</td></tr></table></span>+<span dealflag=\\\"1\\\" class=\\\"MathJye\\\" mathtag=\\\"math\\\" style=\\\"whiteSpace:nowrap;wordSpacing:normal;wordWrap:normal\\\"><table style=\\\"margin-right:1px;vertical-align:text-bottom\\\" cellspacing=\\\"-1\\\" cellpadding=\\\"-1\\\"><tr><td style=\\\"font-size:90%\\\"><div hassize=\\\"9\\\"><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594L.png') repeat-x; width:1px; height:5px;float:left;overflow:hidden\\\" muststretch=\\\"h\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594.png') no-repeat; width:9px; height:5px;float:left;overflow:hidden\\\"></div></div></td></tr><tr><td>b</td></tr></table></span>=t<span dealflag=\\\"1\\\" class=\\\"MathJye\\\" mathtag=\\\"math\\\" style=\\\"whiteSpace:nowrap;wordSpacing:normal;wordWrap:normal\\\"><table style=\\\"margin-right:1px;vertical-align:text-bottom\\\" cellspacing=\\\"-1\\\" cellpadding=\\\"-1\\\"><tr><td style=\\\"font-size:90%\\\"><div hassize=\\\"9\\\"><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594L.png') repeat-x; width:1px; height:5px;float:left;overflow:hidden\\\" muststretch=\\\"h\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594.png') no-repeat; width:9px; height:5px;float:left;overflow:hidden\\\"></div></div></td></tr><tr><td>a</td></tr></table></span>-2t<span dealflag=\\\"1\\\" class=\\\"MathJye\\\" mathtag=\\\"math\\\" style=\\\"whiteSpace:nowrap;wordSpacing:normal;wordWrap:normal\\\"><table style=\\\"margin-right:1px;vertical-align:text-bottom\\\" cellspacing=\\\"-1\\\" cellpadding=\\\"-1\\\"><tr><td style=\\\"font-size:90%\\\"><div hassize=\\\"9\\\"><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594L.png') repeat-x; width:1px; height:5px;float:left;overflow:hidden\\\" muststretch=\\\"h\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/8594.png') no-repeat; width:9px; height:5px;float:left;overflow:hidden\\\"></div></div></td></tr><tr><td>b</td></tr></table></span>，则有<span dealflag=\\\"1\\\" class=\\\"MathJye\\\" mathtag=\\\"math\\\" style=\\\"whiteSpace:nowrap;wordSpacing:normal;wordWrap:normal\\\"><table style=\\\"margin-right: 1px\\\" cellspacing=\\\"-1\\\" cellpadding=\\\"-1\\\"><tr><td style=\\\"font-size: 0\\\"><div hassize=\\\"21\\\"><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/123U.png') no-repeat; width:9px; height:6px;overflow:hidden\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/123M.png') repeat-y; width:9px; height:1px;overflow:hidden\\\" muststretch=\\\"v\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/123L.png') no-repeat; width:9px; height:9px;overflow:hidden\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/123M.png') repeat-y; width:9px; height:1px;overflow:hidden\\\" muststretch=\\\"v\\\"></div><div style=\\\"background: url('//workimg.qiaoxuesi.com/images/formula/part/123D.png') no-repeat; width:9px; height:6px;overflow:hidden\\\"></div></div></td><td><table style=\\\"text-align: left; width: 100%; margin-left: 1px; margin-right: 1px\\\" cellspacing=\\\"-1\\\" cellpadding=\\\"-1\\\"><tr><td>k=t</td></tr><tr><td style=\\\"padding-top:6px\\\">1=-2t</td></tr></table></td><td style=\\\"font-size: 0\\\"></td></tr></table></span>，则k=-<span class=\\\"MathJye\\\" mathtag=\\\"math\\\" style=\\\"whiteSpace:nowrap;wordSpacing:normal;wordWrap:normal\\\"><table cellpadding=\\\"-1\\\" cellspacing=\\\"-1\\\" style=\\\"margin-right:1px\\\"><tr><td style=\\\"border-bottom:1px solid black;padding-bottom:1px;font-size:90%\\\">1</td></tr><tr><td style=\\\"padding-top:1px;font-size:90%\\\">2</td></tr></table></span>；<br />故k=-<span class=\\\"MathJye\\\" mathtag=\\\"math\\\" style=\\\"whiteSpace:nowrap;wordSpacing:normal;wordWrap:normal\\\"><table cellpadding=\\\"-1\\\" cellspacing=\\\"-1\\\" style=\\\"margin-right:1px\\\"><tr><td style=\\\"border-bottom:1px solid black;padding-bottom:1px;font-size:90%\\\">1</td></tr><tr><td style=\\\"padding-top:1px;font-size:90%\\\">2</td></tr></table></span>．\"}]","ocrmath_text":"の $\\vec { a } = ( 1 , 2 ) , \\vec { b } = ( - 3 , 2 ) , \\because 1 \\times 2 \\neq 2 \\times ( - 3 ) , \\therefore \\vec { a } , \\vec { b }$ 不共线\r\n四名 $3 \\vec { a } + 4 \\vec { b } = ( m - 1 ) \\vec { a } + ( 2 - n ) \\vec { b } $, $\\left\\{ \\begin{array} { l } m = 4 \\\\ n = - 2 \\end{array} \\right.$ 2 $\\left\\{ \\begin{array} { l } m + | z 3 \\\\ 2 n - 4 = 1 \\end{array} \\right. ,$ 得\r\n3) \r\n","hh_text":"(1) $\\overrightarrow {a}=(1,2),\\overrightarrow {b}=(-3,2)$  ∵ $1\\times 2\\neq 2\\times (-3),$ ∴a,b不共线\n\n（2）若 $3\\overrightarrow {a}+4\\overrightarrow {b}=(m-1)\\overrightarrow {a}+(2-n)\\overrightarrow {b},$  则 $m+1=3$  $m=4$ 得 $2n-4=1$ $n=-2$ \n\n(3)\n\n"}}