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在长方体 $A B C D - A_(1) B_(1) C_(1) D_(1)$中，下列各式运算结果...

创建于：2026年3月4日

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问题

在长方体 ABCD−A(1)B(1)C(1)D(1)A B C D - A_(1) B_(1) C_(1) D_(1)中，下列各式运算结果为arrow(BD(1))arrow(B D_(1))的是（）

#grid(
columns: (1fr, 1fr),
column-gutter: 2em,
row-gutter: 0.6em,
align: (left, left),
[A. arrow(A(1)D(1))−arrow(A(1)A)−arrow(AB)arrow(A_(1) D_(1))- arrow(A_(1) A)- arrow(A B)],
[B. arrow(BC)+arrow(BB(1))−arrow(D(1)C(1))arrow(B C)+ arrow(B B_(1))- arrow(D_(1) C_(1))],
[C. arrow(AD)−arrow(AB)−arrow(DD(1))arrow(A D)- arrow(A B)- arrow(D D_(1))],
[D. arrow(B(1)D(1))−arrow(A(1)A)+arrow(DD(1))arrow(B_(1) D_(1))- arrow(A_(1) A)+ arrow(D D_(1))],
) 需求：出两道类似题

思考

Verifying vector expressions

We need to determine which expression matches vector BD1 in a rectangular parallelepiped. After carefully calculating each option, I find that options A and B both match, but C doesn’t. Option D also isn’t correct. So, A and B are the correct answers in this case.

Analyzing expressions for vector AC1

I’m checking possibilities to determine which expressions match vector AC1. After evaluating the options, it seems like multiple answers could be correct. Answers A and B both work, but the question may be designed as a multiple-choice, so I’m selecting both. I’ll also create a couple of similar problems to practice.

Designing problems with multiple correct answers

I want to design two problems with only one correct answer each. The target vectors, like AC1 and CB1, will have options that include a correct result. I'll make sure the updates allow for clarity and consistency in the answers. I'll also include a space for "multiple-choice” if needed.

Crafting multiple-choice questions

For the first problem, I’ve crafted four options related to vector AC1, with options A and B matching the target. For the second question about vector DB1, I’ve narrowed it down to A and B. I’ll adjust to make consistent problems, ensuring only exactly two answers are correct.

Thought for 1m 11s

回答

先把长方体设为：AB→,AD→,AA1→\overrightarrow{AB},\overrightarrow{AD},\overrightarrow{AA_1}两两垂直。
则

BD1→=BA→+AD→+DD1→=−AB→+AD→+AA1→.\overrightarrow{BD_1}=\overrightarrow{BA}+\overrightarrow{AD}+\overrightarrow{DD_1} = -\overrightarrow{AB}+\overrightarrow{AD}+\overrightarrow{AA_1}.

逐项看选项：

  • A：A1D1→−A1A→−AB→\overrightarrow{A_1D_1}-\overrightarrow{A_1A}-\overrightarrow{AB}
    其中 A1D1→=AD→\overrightarrow{A_1D_1}=\overrightarrow{AD}，A1A→=−AA1→\overrightarrow{A_1A}=-\overrightarrow{AA_1}，所以

    =AD→−(−AA1→)−AB→=−AB→+AD→+AA1→=BD1→.=\overrightarrow{AD}-(-\overrightarrow{AA_1})-\overrightarrow{AB} =-\overrightarrow{AB}+\overrightarrow{AD}+\overrightarrow{AA_1} =\overrightarrow{BD_1}.

    ✅

  • B：BC→+BB1→−D1C1→\overrightarrow{BC}+\overrightarrow{BB_1}-\overrightarrow{D_1C_1}
    BC→=AD→\overrightarrow{BC}=\overrightarrow{AD}，BB1→=AA1→\overrightarrow{BB_1}=\overrightarrow{AA_1}，D1C1→=AB→\overrightarrow{D_1C_1}=\overrightarrow{AB}，所以

    =AD→+AA1→−AB→=BD1→.=\overrightarrow{AD}+\overrightarrow{AA_1}-\overrightarrow{AB} =\overrightarrow{BD_1}.

    ✅

  • C：结果会变成 −AB→+AD→−AA1→-\overrightarrow{AB}+\overrightarrow{AD}-\overrightarrow{AA_1}，少了方向（zz分量相反）❌

  • D：会多出一段 AA1→\overrightarrow{AA_1}（相当于“竖向”加了两次）❌

所以原题答案：A、B。

类似题 1（多选）

在长方体 ABCD−A1B1C1D1ABCD-A_1B_1C_1D_1 中，下列各式运算结果为 AC1→\overrightarrow{AC_1} 的是（ ）

A. AB→+BC→+CC1→\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CC_1}
B. AD→+DC→+BB1→\overrightarrow{AD}+\overrightarrow{DC}+\overrightarrow{BB_1}
C. A1C1→−A1A→−AB→\overrightarrow{A_1C_1}-\overrightarrow{A_1A}-\overrightarrow{AB}
D. B1C1→−AD→+AA1→\overrightarrow{B_1C_1}-\overrightarrow{AD}+\overrightarrow{AA_1}

答案：A、B。

类似题 2（多选）

在长方体 ABCD−A1B1C1D1ABCD-A_1B_1C_1D_1 中，下列各式运算结果为 DB1→\overrightarrow{DB_1} 的是（ ）

A. DA→+AB→+AA1→\overrightarrow{DA}+\overrightarrow{AB}+\overrightarrow{AA_1}
B. DC→+CC1→−AB→\overrightarrow{DC}+\overrightarrow{CC_1}-\overrightarrow{AB}
C. BD→+BB1→\overrightarrow{BD}+\overrightarrow{BB_1}
D. DD1→+D1A1→+A1B1→\overrightarrow{DD_1}+\overrightarrow{D_1A_1}+\overrightarrow{A_1B_1}

答案：A、D。

如果你想要“单选题版本”（每题只有一个正确选项），我也可以把选项调整得更“坑”。

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