录入2025届高二下学期校本作业-空间向量 新题

工具v2/文本文件/新题收录列表.txt

@@ -367,3 +367,6 @@
 20240224-192316 2025届高二下学期校本作业-解析几何
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+20240224-192835 2025届高二下学期校本作业-空间向量
+022048:022064,022066:022067,022078,022069:022071,041040:041041,022074:022077,022068,022079:022083
+

题库0.3/Problems.json

@@ -750764,5 +750764,45 @@
         "remark": "",
         "space": "4em",
         "unrelated": []
+    },
+    "041040": {
+        "id": "041040",
+        "content": "已知长方体$ABCD-A_1B_1C_1D_1$的上、下底面都是边长为$3$的正方形, 长方体的高为$4$, 如图建立空间直角坐标系. 下列空间向量分别是图中连接哪两个顶点的直线的一个方向向量?\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\def\\l{3}\n\\def\\m{3}\n\\def\\n{4}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below ] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw [dashed] (C)--(A)(D)--(B)($(A)!0.5!(C)$)--($(A1)!0.5!(C1)$);\n\\draw [->] (A) -- ($(A)!-0.5!(C)$) node [below] {$x$} coordinate (x);\n\\draw [->] (B) -- ($(B)!-0.5!(D)$) node [right] {$y$} coordinate (y);\n\\draw [->] ($(A1)!0.5!(C1)$) --++ (0,1.5,0) node [right] {$z$} coordinate (z);\n\\end{tikzpicture}\n\\end{center}\n(1) $\\overrightarrow{a}=(1,0,0)$.\\\\\n(2) $\\overrightarrow{b}=(0,1,0)$.\\\\\n(3) $\\overrightarrow{c}=(3 \\sqrt{2}, 0,4)$.\\\\\n(4) $\\overrightarrow{d}=(0,3 \\sqrt{2}, 8)$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高二下学期校本作业-空间向量",
+        "edit": [
+            "20240224\t蔡海鸾"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "041041": {
+        "id": "041041",
+        "content": "已知长方体$ABCD-A_1B_1C_1D_1$的上、下底面都是边长为$3$的正方形, 长方体的高为$4$, 如图建立空间直角坐标系. 分别求下列平面的一个法向量.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\def\\l{3}\n\\def\\m{3}\n\\def\\n{4}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below ] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw [dashed] (C)--(A)(D)--(B)($(A)!0.5!(C)$)--($(A1)!0.5!(C1)$);\n\\draw [->] (A) -- ($(A)!-0.5!(C)$) node [below] {$x$} coordinate (x);\n\\draw [->] (B) -- ($(B)!-0.5!(D)$) node [right] {$y$} coordinate (y);\n\\draw [->] ($(A1)!0.5!(C1)$) --++ (0,1.5,0) node [right] {$z$} coordinate (z);\n\\end{tikzpicture}\n\\end{center}\n(1) 平面 $AA_1D_1D$.\\\\\n(2) 平面 $BB_1D_1D$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高二下学期校本作业-空间向量",
+        "edit": [
+            "20240224\t蔡海鸾"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
     }
 }