From 710e620f8fc477d0744d146a8fe92cdfc68206d5 Mon Sep 17 00:00:00 2001 From: wangweiye7840 Date: Wed, 5 Jul 2023 15:48:36 +0800 Subject: [PATCH] =?UTF-8?q?=E5=BD=95=E5=85=A5=E7=A9=BA=E4=B8=AD=E8=AF=BE?= =?UTF-8?q?=E5=A0=82=E5=B9=B3=E9=9D=A2=E5=90=91=E9=87=8F=E4=BE=8B=E9=A2=98?= =?UTF-8?q?=E4=B8=8E=E4=B9=A0=E9=A2=98?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 工具v2/批量收录题目.py | 2 +- 题库0.3/Problems.json | 1180 ++++++++++++++++++++++++++++++++++++++++ 2 files changed, 1181 insertions(+), 1 deletion(-) diff --git a/工具v2/批量收录题目.py b/工具v2/批量收录题目.py index 73e8633d..f5bae188 100644 --- a/工具v2/批量收录题目.py +++ b/工具v2/批量收录题目.py @@ -1,5 +1,5 @@ #修改起始id,出处,文件名 -starting_id = 18434 #起始id设置, 来自"寻找空闲题号"功能 +starting_id = 18470 #起始id设置, 来自"寻找空闲题号"功能 raworigin = "" #题目来源的前缀(中缀在.tex文件中) filename = r"C:\Users\wangweiye\Documents\wwy sync\临时工作区\空中课堂必修第二册例题与习题.tex" #题目的来源.tex文件 editor = "王伟叶" #编辑者姓名 diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 1c07a4f7..85e4f255 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -475707,6 +475707,1186 @@ "space": "4em", "unrelated": [] }, + "018470": { + "id": "018470", + "content": "如图, 在等边三角形$ABC$中, $D$、$E$、$F$分别是边$BC$、$AB$、$AC$的中点. 写出图中与向量$\\overrightarrow{EF}$平行的非零向量和与$\\overrightarrow{EF}$相等的向量.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [above] {$A$} coordinate (A);\n\\draw (-60:2) node [right] {$C$} coordinate (C);\n\\draw (-120:2) node [left] {$B$} coordinate (B);\n\\draw ($(A)!0.5!(B)$) node [left] {$E$} coordinate (E);\n\\draw ($(A)!0.5!(C)$) node [right] {$F$} coordinate (F);\n\\draw ($(B)!0.5!(C)$) node [below] {$D$} coordinate (D);\n\\draw (A)--(B)--(C)--cycle(D)--(E)--(F)--cycle;\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018471": { + "id": "018471", + "content": "如图, 在等边三角形$ABC$中, $D$、$E$、$F$分别是边$BC$、$AB$、$AC$的中点.写出向量$\\overrightarrow{AE}$的负向量.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [above] {$A$} coordinate (A);\n\\draw (-60:2) node [right] {$C$} coordinate (C);\n\\draw (-120:2) node [left] {$B$} coordinate (B);\n\\draw ($(A)!0.5!(B)$) node [left] {$E$} coordinate (E);\n\\draw ($(A)!0.5!(C)$) node [right] {$F$} coordinate (F);\n\\draw ($(B)!0.5!(C)$) node [below] {$D$} coordinate (D);\n\\draw (A)--(B)--(C)--cycle(D)--(E)--(F)--cycle;\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018472": { + "id": "018472", + "content": "已知向量$\\overrightarrow {a}$与$\\overrightarrow {b}$不平行, 若$\\overrightarrow {c}\\parallel \\overrightarrow {a}$且$\\overrightarrow {c}\\parallel \\overrightarrow {b}$, 则下列说法正确的是\\bracket{20}.\n\\twoch{$\\overrightarrow {c}=\\overrightarrow{0}$}{$\\overrightarrow {c}=\\overrightarrow {a}$}{$\\overrightarrow {c}=\\overrightarrow {b}$}{不存在满足条件的向量$\\overrightarrow {c}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018473": { + "id": "018473", + "content": "判断下列命题的真假, 并说明理由:\\\\\n(1) 若$\\overrightarrow {a}$与$\\overrightarrow {b}$都是单位向量, 则$\\overrightarrow {a}=\\overrightarrow {b}$;\\\\\n(2) 方向为南偏西$60^{\\circ}$的向量与北偏东$60^{\\circ}$的向量是平行向量;\\\\\n(3) 若$\\overrightarrow {a}$与$\\overrightarrow {b}$是平行向量, 则$\\overrightarrow {a}=\\overrightarrow {b}$;\\\\\n(4) 若向量$\\overrightarrow{AM}$与$\\overrightarrow{AN}$不相等, 则点$M$与$N$不重合.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018474": { + "id": "018474", + "content": "判断下列命题真假, 并说明理由:\\\\\n(1) 若$|\\overrightarrow {a}|=|\\overrightarrow {b}|$, $|\\overrightarrow {b}|=|\\overrightarrow {c}|$, 则$|\\overrightarrow {a}|=|\\overrightarrow {c}|$;\\\\\n(2) 若$\\overrightarrow {a}\\parallel \\overrightarrow {b}, \\overrightarrow {b}\\parallel \\overrightarrow {c}$, 则$\\overrightarrow {a}\\parallel \\overrightarrow {c}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018475": { + "id": "018475", + "content": "已知点$O$是矩形$ABCD$的对角线$AC$与$BD$的交点, 设点集$M=\\{A, B, C, D, O\\}$, 向量的集合$T=\\{\\overrightarrow{PQ} | P, Q \\in M$, 且$P$、$Q$不重合$\\}$. 试求集合$T$的元素个数.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018476": { + "id": "018476", + "content": "物体受水平方向$6 \\mathrm{N}$和铅垂方向$8 \\mathrm{N}$的两个力的作用, 求合力的大小以及合力与铅垂方向偏离的角度. (结果精确到$0.01^{\\circ}$)", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018477": { + "id": "018477", + "content": "已知$\\triangle ABC$是边长为$1$的等边三角形, 点$O$是$\\triangle ABC$所在平面上的任意一点.求向量$(\\overrightarrow{OA}-\\overrightarrow{OC})+(\\overrightarrow{OB}-\\overrightarrow{OC})$的模.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018478": { + "id": "018478", + "content": "若点$O$是$\\triangle ABC$所在平面内的一点, 且满足$|\\overrightarrow{OB}-\\overrightarrow{OC}|=|(\\overrightarrow{OB}-\\overrightarrow{OA})+(\\overrightarrow{OC}-\\overrightarrow{OA})|$, 则$\\triangle ABC$的形状为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018479": { + "id": "018479", + "content": "巳知$|\\overrightarrow {a}|=2$, $|\\overrightarrow {b}|=1$, $|\\overrightarrow {a}-\\overrightarrow {b}|=\\sqrt{3}$, 求$|\\overrightarrow {a}+\\overrightarrow {b}|$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018480": { + "id": "018480", + "content": "化简下列向量线性运算:\\\\\n(1) $(-2) \\times(\\dfrac{1}{2} \\overrightarrow {a})$;\\\\\n(2) $2(\\overrightarrow {a}-\\overrightarrow {b})+3(\\overrightarrow {a}+\\overrightarrow {b})$;\\\\\n(3) $(\\lambda-\\mu)(\\overrightarrow {a}+\\overrightarrow {b})-(\\lambda+\\mu)(\\overrightarrow {a}-\\overrightarrow {b})$.($\\lambda, \\mu \\in \\mathbf{R}$)", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018481": { + "id": "018481", + "content": "已知向量$\\overrightarrow {a}$、$\\overrightarrow {b}$、$\\overrightarrow {c}$满足$\\dfrac{1}{2}(\\overrightarrow {a}-3 \\overrightarrow {c})+2(2 \\overrightarrow {a}-3 \\overrightarrow {b})=\\overrightarrow{0}$, 试用$\\overrightarrow {a}$、$\\overrightarrow {b}$表示$\\overrightarrow {c}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018482": { + "id": "018482", + "content": "如图, 在$\\triangle ABC$中, $D$是$AB$的中点, $E$是$BC$延长线上一点, 且$BE=2BC$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$B$} coordinate (B);\n\\draw (1.2,0.2) node [below] {$C$} coordinate (C);\n\\draw (1,1.2) node [above] {$A$} coordinate (A);\n\\draw ($(A)!0.5!(B)$) node [above left] {$D$} coordinate (D);\n\\draw ($(B)!2!(C)$) node [right] {$E$} coordinate (E);\n\\draw (A)--(B)--(E)--(D)(A)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 用向量$\\overrightarrow{BA}$、$\\overrightarrow{BC}$表示$\\overrightarrow{DE}$;\\\\\n(2) 用向量$\\overrightarrow{CA}$、$\\overrightarrow{CB}$表示$\\overrightarrow{DE}$;", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018483": { + "id": "018483", + "content": "已知任意两个非零向量$\\overrightarrow {a}$和$\\overrightarrow {b}$, 试作$\\overrightarrow{OA}=\\overrightarrow {a}+\\overrightarrow {b}$, $\\overrightarrow{OB}=\\overrightarrow {a}+2 \\overrightarrow {b}$, $\\overrightarrow{OC}=\\overrightarrow {a}+3 \\overrightarrow {b}$. 判断$A$、$B$、$C$三点之间的位置关系, 并证明你的猜想.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018484": { + "id": "018484", + "content": "如图, 在$\\triangle ABC$中, $\\overrightarrow{BC}=3 \\overrightarrow{BD}$, $\\overrightarrow{AE}=\\dfrac{2}{3} \\overrightarrow{AD}$, $\\overrightarrow{CE}=\\lambda \\overrightarrow{AB}+\\mu \\overrightarrow{AC}$, 求实数$\\lambda$和$\\mu$的值.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$B$} coordinate (B);\n\\draw (0.9,0.1) node [below] {$D$} coordinate (D);\n\\draw (1,1.2) node [above] {$A$} coordinate (A);\n\\draw ($(B)!3!(D)$) node [right] {$C$} coordinate (C);\n\\draw ($(A)!{2/3}!(D)$) node [left] {$E$} coordinate (E);\n\\draw (A)--(B)--(C)--cycle(A)--(D)(E)--(C);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018485": { + "id": "018485", + "content": "已知$\\overrightarrow {a}$和$\\overrightarrow {b}$是两个不平行的向量, 向量$\\overrightarrow {b}-t \\overrightarrow {a}$与$\\dfrac{1}{2} \\overrightarrow {a}-\\dfrac{3}{2} \\overrightarrow {b}$平行, 求实数$t$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018486": { + "id": "018486", + "content": "已知向量$\\overrightarrow {a}$与$\\overrightarrow {b}$的夹角为$\\dfrac{2 \\pi}{3}$, 且$|\\overrightarrow {a}|=3$, $|\\overrightarrow {b}|=4$.\\\\\n(1) 求$\\overrightarrow {b}$在$\\overrightarrow {a}$方向上的投影与数量投影;\\\\\n(2) 求$\\overrightarrow {b} \\cdot \\overrightarrow {a}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018487": { + "id": "018487", + "content": "已知$\\overrightarrow {a}$是非零向量, $\\overrightarrow {b}$与$\\overrightarrow {c}$是任意向量, 它们在$\\overrightarrow {a}$方向上的投影分别为$\\overrightarrow{b'}$与$\\overrightarrow{c'}$. 求证: $\\overrightarrow {b}+\\overrightarrow {c}$在$\\overrightarrow {a}$方向上的投影为$\\overrightarrow{b'}+\\overrightarrow{c'}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018488": { + "id": "018488", + "content": "已知圆$O$中, 弦$AB$的长为$\\sqrt{3}$, 圆上的点$C$满足$\\overrightarrow{OA}+\\overrightarrow{OB}+\\overrightarrow{OC}=\\overrightarrow{0}$, 求$\\overrightarrow{AC}$在$\\overrightarrow{OA}$方向上的数量投影.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018489": { + "id": "018489", + "content": "如图, 已知正三角形$ABC, O$是三角形$ABC$外一点, 且$|OA|=1$, 若$\\angle OAB=\\theta$($\\theta \\in(0, \\dfrac{\\pi}{3})$), 试用向量的方法证明: $\\cos \\theta+\\cos (\\theta+\\dfrac{2 \\pi}{3})+\\cos (\\theta+\\dfrac{4 \\pi}{3})=0$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [above] {$A$} coordinate (A);\n\\draw (-60:1.5) node [right] {$C$} coordinate (C);\n\\draw (-120:1.5) node [left] {$B$} coordinate (B);\n\\draw (A) --++ (-160:1) node [left] {$O$} coordinate (O);\n\\draw pic [draw, \"$\\theta$\", scale = 0.6, angle eccentricity = 2.5] {angle = O--A--B};\n\\draw (A)--(B)--(C)--cycle;\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018490": { + "id": "018490", + "content": "证明:\\\\\n(1) $(\\overrightarrow {a}+\\overrightarrow {b})^2=\\overrightarrow {a}^2+2 \\overrightarrow {a} \\cdot \\overrightarrow {b}+\\overrightarrow {b}^2$;\\\\\n(2) $(\\overrightarrow {a}+\\overrightarrow {b}) \\cdot(\\overrightarrow {a}-\\overrightarrow {b})=\\overrightarrow {a}^2-\\overrightarrow {b}^2$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018491": { + "id": "018491", + "content": "设向量$\\overrightarrow {a}$、$\\overrightarrow {b}$满足$|\\overrightarrow {a}|=2$, $|\\overrightarrow {b}|=3$, $\\langle\\overrightarrow {a}, \\overrightarrow {b}\\rangle=\\dfrac{\\pi}{3}$. 求$|3 \\overrightarrow {a}-2 \\overrightarrow {b}|$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018492": { + "id": "018492", + "content": "已知向量$\\overrightarrow {a}$、$\\overrightarrow {b}$满足$|\\overrightarrow {a}|=4$, $\\overrightarrow {b}$在$\\overrightarrow {a}$方向上的数量投影为 $-2$ , 求$|\\overrightarrow {a}-3 \\overrightarrow {b}|$的最小值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018493": { + "id": "018493", + "content": "如图, $O$是线段$AB$外一点, $|OA|=3$, $|OB|=2$, $P$是线段$AB$的垂直平分线$l$上的动点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\filldraw (0,0) circle (0.03) node [left] {$B$} coordinate (B);\n\\filldraw (3,2) circle (0.03) node [right] {$A$} coordinate (A);\n\\draw ($(A)!0.5!(B)$) ++ (-1,1.5) node [below] {$l$} --++ (2,-3) ++ (-0.2,0.3) node [right] {$P$} coordinate (P);\n\\filldraw (P) circle (0.03);\n\\filldraw ({24/13},{-10/13}) node [below] {$O$} coordinate (O);\n\\draw (O)--(P)(A)--(B)--(O)--cycle;\n\\end{tikzpicture}\n\\end{center}\n(1) 求$\\overrightarrow{AB} \\cdot(\\overrightarrow{OA}+\\overrightarrow{OB})$;\\\\\n(2) 求$\\overrightarrow{OP} \\cdot \\overrightarrow{AB}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018494": { + "id": "018494", + "content": "如图, 在平行四边形$ABCD$中, 两条对角线的交点是$M$, 设$\\overrightarrow{AB}=\\overrightarrow {a}$, $\\overrightarrow{AD}=\\overrightarrow {b}$. 试用$\\overrightarrow {a}$、$\\overrightarrow {b}$的线性组合分别表示$\\overrightarrow{MA}$、$\\overrightarrow{MB}$、$\\overrightarrow{MC}$与$\\overrightarrow{MD}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$A$} coordinate (A);\n\\draw (2.5,0) node [below] {$B$} coordinate (B);\n\\draw (1,2) node [above] {$D$} coordinate (D);\n\\draw (3.5,2) node [above] {$C$} coordinate (C);\n\\draw [->] (A)--(B) node [midway, below] {$\\overrightarrow{a}$};\n\\draw [->] (A)--(D) node [midway, left] {$\\overrightarrow{b}$};\n\\draw (A)--(C)(B)--(D)--(C)--cycle;\n\\draw ($(A)!0.5!(C)$) node [above] {$M$} coordinate (M);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018495": { + "id": "018495", + "content": "设$\\overrightarrow {a}$、$\\overrightarrow {b}$是平面内不平行的两个向量, 实数$\\lambda$、$\\mu$满足$3 \\lambda \\overrightarrow {a}+(10-\\mu) \\overrightarrow {b}=(2 \\mu+1) \\overrightarrow {a}+2 \\lambda \\overrightarrow {b}$, 试求$\\lambda$与$\\mu$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018496": { + "id": "018496", + "content": "已知$\\overrightarrow {a}$、$\\overrightarrow {b}$、$\\overrightarrow {c}$是平面上任意三个给定的向量, 其中$\\overrightarrow {a}$和$\\overrightarrow {b}$不平行, 将满足$x^2 \\overrightarrow {a}+x \\overrightarrow {b}+\\overrightarrow {c}=\\overrightarrow{0}$的实数$x$的个数记为$n$, 则$n$的值的集合为\\bracket{20}.\n\\fourch{$\\{0\\}$}{$\\{1\\}$}{$\\{2\\}$}{$\\{0,1\\}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018497": { + "id": "018497", + "content": "如图, 平面上三个向量$\\overrightarrow{OA}$、$\\overrightarrow{OB}$、$\\overrightarrow{OC}$满足: $|\\overrightarrow{OA}|=2,|\\overrightarrow{OB}|=\\sqrt{3}$, $|\\overrightarrow{OC}|=1$, 且$\\angle AOB=120^{\\circ}$, $\\angle AOC=150^{\\circ}$, 设$\\overrightarrow{OC}=m \\overrightarrow{OA}+n \\overrightarrow{OB}$, 求实数$m$、$n$的值.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$O$} coordinate (O);\n\\draw (O) ++ (2,0) node [right] {$A$} coordinate (A);\n\\draw (O) ++ (120:{sqrt(3)}) node [left] {$B$} coordinate (B);\n\\draw (O) ++ (-150:1) node [left] {$C$} coordinate (C);\n\\foreach \\i in {A,B,C}\n{\\draw [->] (O) -- (\\i);};\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018498": { + "id": "018498", + "content": "如图, 写出向量$\\overrightarrow {a}$、$\\overrightarrow {b}$与$\\overrightarrow {c}$的坐标.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2.5,0) -- (2.5,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {-2,-1,1,2}\n{\\draw [gray, dashed] (-2,\\i) -- (2,\\i) (\\i,-2) -- (\\i,2);\n\\draw (\\i,0.2) -- (\\i,0) node [below] {$\\i$} (0.2,\\i) -- (0,\\i) node [left] {$\\i$};};\n\\draw [->] (0,0) -- (1,2) node [midway, below right] {$\\overrightarrow{a}$};\n\\draw [->] (1,-2) -- (2,0) node [midway, below right] {$\\overrightarrow{b}$};\n\\draw [->] (-1,0) -- (0,-2) node [midway, below left] {$\\overrightarrow{c}$};\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018499": { + "id": "018499", + "content": "给定向量$\\overrightarrow {a}=(4,-1)$与$\\overrightarrow {b}=(5,2)$, 求向量$2 \\overrightarrow {a}+3 \\overrightarrow {b}$的坐标.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018500": { + "id": "018500", + "content": "平面上$A$、$B$、$C$三点的坐标分别为$(2,1)$、$(-3,2)$、$(-1,3)$, 写出向量$\\overrightarrow{AC}$、$\\overrightarrow{BC}$的坐标.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018501": { + "id": "018501", + "content": "已知平面内两点$P$、$Q$的坐标分别为$(-2,4)$、$(2,1)$, 求$\\overrightarrow{PQ}$的单位向量$\\overrightarrow{a_0}$的坐标.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018502": { + "id": "018502", + "content": "已知平面上$A$、$B$、$C$三点的坐标分别为$(-1,-1)$、$(2,-7)$、$(-3,3)$, 请尝试判断$A$、$B$、$C$三点是否共线.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018503": { + "id": "018503", + "content": "已知$A$、$B$、$C$、$D$是一个平行四边形的四个顶点, 若$A$、$B$、$C$的坐标分别为$(-2,1)$、$(-1,3)$、$(3,4)$, 求顶点$D$的坐标.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018504": { + "id": "018504", + "content": "若$\\overrightarrow{a_1}, \\overrightarrow{a_2}, \\overrightarrow{a_3}$均为单位向量, 则$\\overrightarrow{a_1}=(\\dfrac{\\sqrt{3}}{3}$, $\\dfrac{\\sqrt{6}}{3})$是$\\overrightarrow{a_1}+\\overrightarrow{a_2}+\\overrightarrow{a_3}=(\\sqrt{3}, \\sqrt{6})$的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018505": { + "id": "018505", + "content": "已知向量$\\overrightarrow {a}=(1,2)$, $\\overrightarrow {b}=(2,-2)$. 求$|\\overrightarrow {a}|$、$|\\overrightarrow {b}|$与$\\langle\\overrightarrow {a}, \\overrightarrow {b}\\rangle$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018506": { + "id": "018506", + "content": "已知$\\triangle ABC$中$A$、$B$、$C$三点的坐标分别为$(2,-2)$、$(-2,3)$、$(3,7)$, 求证: $\\triangle ABC$为直角三角形.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018507": { + "id": "018507", + "content": "已知$x_1$、$x_2$、$y_1$、$y_2$都是实数, 求证: $(x_1 x_2+y_1 y_2)^2 \\leq(x_1^2+y_1^2)(x_2^2+y_2^2)$, 并且等号成立的一个充要条件是$x_1 y_2=x_2 y_1$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018508": { + "id": "018508", + "content": "设$x$、$y \\in \\mathbf{R}$, 向量$\\overrightarrow {a}=(x, 1)$, $\\overrightarrow {b}=(1, y)$, $\\overrightarrow {c}=(2,-4)$, 且$\\overrightarrow {a} \\perp \\overrightarrow {c}$, $\\overrightarrow {b}\\parallel \\overrightarrow {c}$, 则$|\\overrightarrow {a}+\\overrightarrow {b}|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018509": { + "id": "018509", + "content": "已知两个向量$\\overrightarrow {a}$和$\\overrightarrow {b}$满足$\\overrightarrow {a}+\\overrightarrow {b}=(2,-8)$, $\\overrightarrow {a}-\\overrightarrow {b}=(-6,-4)$, 求$\\overrightarrow {a}$与$\\overrightarrow {b}$的夹角.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018510": { + "id": "018510", + "content": "证明对角线互相平分的四边形是平行四边形.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018511": { + "id": "018511", + "content": "已知$P$是直线$P_1P_2$上一点, 且$\\overrightarrow{P_1P}=\\lambda \\overrightarrow{PP_2}$($\\lambda$为实数, 且$\\lambda \\neq-1$), $P_1$、$P_2$的坐标分别为$(x_1, y_1)$、$(x_2, y_2)$, 求点$P$的坐标$(x, y)$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018512": { + "id": "018512", + "content": "已知$\\triangle ABC$的三个顶点$A$、$B$、$C$的坐标分别是$(x_1, y_1) 、(x_2, y_2)$、$(x_3, y_3)$, 求此三角形重心$G$的坐标.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018513": { + "id": "018513", + "content": "在$\\triangle ABC$中, 设$\\overrightarrow{CA}=\\overrightarrow {a}, \\overrightarrow{CB}=\\overrightarrow {b}$, 记$\\triangle ABC$的面积为$S$\n(1) 求证: $S=\\dfrac{1}{2} \\sqrt{|\\overrightarrow {a}|^2|\\overrightarrow {b}|^2-(\\overrightarrow {a} \\cdot \\overrightarrow {b})^2}$;\n(2)设$\\overrightarrow {a}=(x_1, y_1), \\overrightarrow {b}=(x_2, y_2)$, 求证: $S=\\dfrac{1}{2}|x_1 y_2-x_2 y_1|$.\n$\\overrightarrow{OC} \\perp$. 设$OA$、$B$、$C$是平面上四点, 已知$\\overrightarrow{OA} \\perp \\overrightarrow{BC}, \\overrightarrow{OB} \\perp \\overrightarrow{AC}$. 求证:", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018514": { + "id": "018514", + "content": "求证: 两组对边平方和相等的四边形的对角线互相垂直.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018515": { + "id": "018515", + "content": "用向量方法证明: $\\cos (\\alpha-\\beta)=\\cos \\alpha \\cos \\beta+\\sin \\alpha \\sin \\beta$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018516": { + "id": "018516", + "content": "如图, 平面上$A, B, C$三点的坐标分别是$(2,3)$、$(2,0)$、$(1,1)$, 已知小明在点$B$处休憩, 有只小狗沿着$AC$所在直线来回跑动. 问: 其在什么位置时, 离小明最近?\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw [->] (-0.5,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (2,0) node [below] {$B$} coordinate (B);\n\\draw (2,3) node [above] {$A$} coordinate (A);\n\\draw (1,1) node [left] {$C$} coordinate (C);\n\\draw (A) -- ($(A)!1.6!(C)$) (C)--(B)--(A);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018517": { + "id": "018517", + "content": "将质量为$20 \\mathrm{kg}$的物体用两根绳子悬挂起来, 如图, 两根绳子与铅锤线的夹角分别为$45^{\\circ}$与$30^{\\circ}$, 求它们分别提供的拉力的大小. (结果精确到$0.1 \\mathrm{N}$)\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) coordinate (O);\n\\draw (O) ++ ({-2/sqrt(3)},2) coordinate (S) (O) ++ (2,2) coordinate (T);\n\\draw ($(S)!-0.2!(T)$) -- ($(T)!-0.2!(S)$);\n\\draw (S)--(O)--(T);\n\\draw [->] (O) --++ (120:2) node [left] {$\\overrightarrow{f_1}$};\n\\draw [->] (O) --++ (1,1) node [right] {$\\overrightarrow{f_2}$};\n\\draw [dashed] (O) --++ (0,2) coordinate (P);\n\\draw pic [\"$30^\\circ$\", angle eccentricity = 2.5] {angle = P--O--S};\n\\draw pic [\"$45^\\circ$\", angle eccentricity = 2] {angle = T--O--P};\n\\draw (O) -- (0,-0.5) coordinate (A);\n\\draw (A) ++ (-0.3,-0.6) rectangle++ (0.6,0.6);\n\\draw (A) ++ (0.3,-0.3) node [right] {$20$kg};\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018518": { + "id": "018518", + "content": "某人朝正南方向游去, 他在静水中游泳速度的大小是$\\sqrt{3} \\mathrm{m} / \\mathrm{s}$, 河水自西向东流速为$1 \\mathrm{m} / \\mathrm{s}$, 求他实际前进的速度的大小和方向.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018519": { + "id": "018519", + "content": "已知向量$\\overrightarrow {a}, \\overrightarrow {b}, \\overrightarrow {c}, \\overrightarrow {d}$, 且$\\overrightarrow {a}, \\overrightarrow {b}$中至少有一个不是零向量, 实数$t$的函数$y=|t \\overrightarrow {a}+\\overrightarrow {c}|^2+|t \\overrightarrow {b}+\\overrightarrow {d}|^2$, 当$t=$\\blank{50}时, $y$取得最小值.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018520": { + "id": "018520", + "content": "已知点$A(3,0)$, $B(0,3)$, $C(\\cos \\alpha, \\sin \\alpha)$, $\\alpha \\in(\\dfrac{\\pi}{2}, \\dfrac{3 \\pi}{2})$. 若$\\overrightarrow{AC} \\cdot \\overrightarrow{BC}=-1$, 求$\\dfrac{2 \\sin ^2 \\alpha+\\sin 2 \\alpha}{1+\\tan \\alpha}$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018521": { + "id": "018521", + "content": "在平行四边形$ABCD$中, $\\angle A=\\dfrac{\\pi}{3}$, 边$AB$、$AD$的长分别为 $2$、$1$. 若$M$、$N$分别是边$BC$、$CD$上的点, 且满足$\\dfrac{|\\overrightarrow{BM}|}{|\\overrightarrow{BC}|}=\\dfrac{|\\overrightarrow{CN}|}{|\\overrightarrow{CD}|}$, 则$\\overrightarrow{AM} \\cdot \\overrightarrow{AN}$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018522": { + "id": "018522", + "content": "用向量的方法证明三角形的正弦定理.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018523": { + "id": "018523", + "content": "如图, $OM\\parallel AB$, 点$P$在由射线$OM$、 线段$OB$及$AB$的延长线围成的阴影区域内 (不含边界) 运动, 且$\\overrightarrow{OP}=x \\overrightarrow{OA}+y \\overrightarrow{OB}$($x, y \\in \\mathbf{R}$). 在阴影区域内作$BF\\parallel OA$, 使得$OABF$构成平行四边形, 当$x=-\\dfrac{1}{2}$, $y=1$时, 点$P$的位置在\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (2,0) node [below] {$A$} coordinate (A);\n\\draw (1.1,1.5) node [above right] {$B$} coordinate (B);\n\\draw ($(A)!1.7!(B)$) coordinate (T);\n\\draw ($(O) + 1.2*(B) - 1.2*(A)$) node [above] {$M$} coordinate (M);\n\\fill [pattern = north east lines] (M)--(T)--(B)--(O)--cycle;\n\\draw [dashed] (B)--(T) (O)--(M);\n\\draw [->] (O)--(A);\n\\draw [->] (O)--(B);\n\\draw (A)--(B);\n\\draw [->] (O) --++ (-0.1,0.6) node [above, fill = white] {$P$} coordinate (P); \n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\triangle OBF$的重心}{线段$OF$的中点}{线段$OB$的中点}{线段$BF$的中点}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018524": { + "id": "018524", + "content": "如图, $OM\\parallel AB$, 点$P$在由射线$OM$、线段$OB$及$AB$的延长线围成的阴影区域内 (不含边界)运动, 且$\\overrightarrow{OP}=x \\overrightarrow{OA}+y \\overrightarrow{OB}$($x, y \\in \\mathbf{R}$). 当$x=-\\dfrac{1}{2}$时, $y$的取值范围是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (2,0) node [below] {$A$} coordinate (A);\n\\draw (1.1,1.5) node [above right] {$B$} coordinate (B);\n\\draw ($(A)!1.7!(B)$) coordinate (T);\n\\draw ($(O) + 1.2*(B) - 1.2*(A)$) node [above] {$M$} coordinate (M);\n\\fill [pattern = north east lines] (M)--(T)--(B)--(O)--cycle;\n\\draw [dashed] (B)--(T) (O)--(M);\n\\draw [->] (O)--(A);\n\\draw [->] (O)--(B);\n\\draw (A)--(B);\n\\draw [->] (O) --++ (-0.1,0.6) node [above, fill = white] {$P$} coordinate (P); \n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018525": { + "id": "018525", + "content": "如图, 在$\\triangle ABC$中, 点$M$是边$BC$的中点, 点$N$是边$AC$的中点, $AM$与$BN$交于点$P$, 用向量的方法求$\\dfrac{AP}{AM}$的值.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$B$} coordinate (B);\n\\draw (2,0) node [right] {$C$} coordinate (C);\n\\draw (1.2,1.6) node [above] {$A$} coordinate (A);\n\\draw ($(A)!0.5!(C)$) node [above right] {$N$} coordinate (N);\n\\draw ($(B)!0.5!(C)$) node [below] {$M$} coordinate (M);\n\\draw ($(B)!{2/3}!(N)$) node [above left] {$P$} coordinate (P);\n\\draw (A)--(B)--(C)--cycle(A)--(M)(B)--(N);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018526": { + "id": "018526", + "content": "如图, 在一个平面内, $\\triangle ABC$为直角三角形, $A$为直角, $AB=3$, $BC=4$, 长为$10$的线段$PQ$以点$A$为中点, 当$\\overrightarrow{PQ}$与$\\overrightarrow{BC}$的夹角$\\theta$取何值时, $\\overrightarrow{BP} \\cdot \\overrightarrow{CQ}$的值最大? 并求出这个最大值.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.3]\n\\draw (0,0) node [below] {$A$} coordinate (A);\n\\draw (3,0) node [right] {$B$} coordinate (B);\n\\draw (0,4) node [above] {$C$} coordinate (C);\n\\draw (40:5) node [above] {$P$} coordinate (P);\n\\draw (220:5) node [below] {$Q$} coordinate (Q);\n\\draw (P)--(Q)(A)--(B)--(C)--cycle;\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018527": { + "id": "018527", + "content": "已知正三角形$ABC$边长为$2$, 圆$A$的半径是$1, PQ$为圆$A$的任意一条直径.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [above] {$A$} coordinate (A);\n\\draw (-60:2) node [right] {$C$} coordinate (C);\n\\draw (-120:2) node [left] {$B$} coordinate (B);\n\\draw (A) circle (1)(A)--(B)--(C)--cycle;\n\\draw (-10:1) node [right] {$Q$} coordinate (Q);\n\\draw (170:1) node [left] {$P$} coordinate (P);\n\\draw (P)--(Q);\n\\end{tikzpicture}\n\\end{center}\n(1) 判断$\\overrightarrow{BP} \\cdot \\overrightarrow{CQ}-\\overrightarrow{AP} \\cdot \\overrightarrow{CB}$是否为定值, 请说明理由;\\\\\n(2) 求$\\overrightarrow{BP} \\cdot \\overrightarrow{CQ}$的最大值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018528": { + "id": "018528", + "content": "已知向量$\\overrightarrow{OA}=(1,7)$, $\\overrightarrow{OB}=(5,1)$, $\\overrightarrow{OP}=(2,1)$, $K$为直线$OP$上的一个动点, 当$\\overrightarrow{KA} \\cdot \\overrightarrow{KB}$取最小值时, 求向量$\\overrightarrow{OK}$的坐标.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂必修第二册平面向量例题与习题", + "edit": [ + "20230705\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, "020001": { "id": "020001", "content": "判断下列各组对象能否组成集合, 若能组成集合, 指出是有限集还是无限集.\\\\\n(1) 上海市控江中学$2022$年入学的全体高一年级新生;\\\\\n(2) 中国现有各省的名称;\\\\\n(3) 太阳、$2$、上海市;\\\\\n(4) 大于$10$且小于$15$的有理数;\\\\\n(5) 末位是$3$的自然数;\\\\\n(6) 影响力比较大的中国数学家;\\\\\n(7) 方程$x^2+x-3=0$的所有实数解;\\\\ \n(8) 函数$y=\\dfrac 1x$图像上所有的点;\\\\ \n(9) 在平面直角坐标系中, 到定点$(0, 0)$的距离等于$1$的所有点;\\\\\n(10) 不等式$3x-10<0$的所有正整数解;\\\\\n(11) 所有的平面四边形.",