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Giải bài 1 trang 60 sách bài tập toán 12 - Cánh diều

Cho tứ diện (ABCD). Lấy (G) là trọng tâm tam giác (BCD). Phát biểu nào sau đây là sai? A. (overrightarrow {GB} + overrightarrow {GC} + overrightarrow {GD} = overrightarrow 0 ). B. (overrightarrow {GA} + overrightarrow {GB} + overrightarrow {GC} + overrightarrow {GD} = overrightarrow 0 ). C. (overrightarrow {CB} + overrightarrow {CD} = 3overrightarrow {CG} ). D. (overrightarrow {AB} + overrightarrow {AC} + overrightarrow {AD} = 3overrightarrow {AG} ).

♌Tổng hợp đề thi học kì 2 lớp 12 tất cả các môn - Cánh diều

Toán - Văn - Anh - Hoá - Sinh - Sử - Địa
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Đề bài

Cho tứ diện \(ABCD\). Lấy \(G\) là trọng tâm tam giác \(BCD\). Phát biểu nào sau đây là sai?

A.ꦏ \(\overrightarrow {GB}  + \overrightarrow {GC}  + \overrightarrow {GD}  = \overrightarrow 0 \)

B.🔴 \(\overrightarrow {GA}  + \overrightarrow {GB}  + \overrightarrow {GC}  + \overrightarrow {GD}  = \overrightarrow 0 \)

C.﷽ \(\overrightarrow {CB}  + \overrightarrow {CD}  = 3\overrightarrow {CG} \)

D.☂ \(\overrightarrow {AB}  + \overrightarrow {AC}  + \overrightarrow {AD}  = 3\overrightarrow {AG} \).

Phương pháp giải - Xem chi tiết

‒ Sử dụng tính chất trung điểm của đoạn thẳng và tính chất trọng tâm của tam giác.

Lời giải chi tiết

Theo tính chất trọng tâm của tam giác, với \(G\) là trọng tâm tam giác \(BCD\), ta có: \(\overrightarrow {GB}  + \overrightarrow {GC}  + \overrightarrow {GD}  = \overrightarrow 0 \). Vậy A đúng. Gọi \(M\) là trung điểm của \(BD\). Ta có: \(\overrightarrow {CB}  + \overrightarrow {CD}  = 2\overrightarrow {CM}  = 2.\frac{2}{3}\overrightarrow {CG}  = 3\overrightarrow {CG} \). Vậy C đúng. Ta có: \(\overrightarrow {AB}  + \overrightarrow {AC}  + \overrightarrow {AD}  = \overrightarrow {AG}  + \overrightarrow {GB}  + \overrightarrow {AG}  + \overrightarrow {GC}  + \overrightarrow {AG}  + \overrightarrow {GD}  = 3\overrightarrow {AG}  + \left( {\overrightarrow {GB}  + \overrightarrow {GC}  + \overrightarrow {GD} } \right) = 3\overrightarrow {AG} \). Vậy D đúng. Chọn B.

  • 🐷 Giải bài 2 trang 60 sách bài tập toán 12 - Cánh diều Cho hình hộp (ABCD.A'B'C'D'). Phát biểu nào nào sau đây là đúng? A. (overrightarrow {AB} + overrightarrow {AD} + overrightarrow {BB'} = overrightarrow {AC'} ). B. (overrightarrow {A'B'} + overrightarrow {A'D'} + overrightarrow {A'A} = overrightarrow {AC'} ). C. (overrightarrow {AB} + overrightarrow {BD} + overrightarrow {A'A} = overrightarrow {AC'} ). D. (overrightarrow {AB} + overrightarrow {AD} + overrightarrow {A'A} = overrightarrow {AC'} ).
  • ✨ Giải bài 3 trang 60 sách bài tập toán 12 - Cánh diều Phát biểu nào nào sau đây là đúng? A. Với hai vectơ bất kì (overrightarrow a ,overrightarrow b ) và số thực (k), ta có: (kleft( {overrightarrow a + overrightarrow b } right) = koverrightarrow a + koverrightarrow b ). B. Với hai vectơ bất kì (overrightarrow a ,overrightarrow b ) và số thực (k), ta có: (kleft( {overrightarrow a + overrightarrow b } right) = overrightarrow a k + overrightarrow b k). C. Với hai vectơ bất kì (overrightarrow a ,overrightarrow b )
  • 🌳 Giải bài 4 trang 60 sách bài tập toán 12 - Cánh diều Cho hình lập phương (ABCD.A'B'C'D'). Góc giữa hai vectơ (overrightarrow {BD} ,overrightarrow {B'C} ) bằng: A. ({30^ circ }). B. ({45^ circ }). C. ({120^ circ }). D. ({60^ circ }).
  • 🦂 Giải bài 5 trang 60 sách bài tập toán 12 - Cánh diều Cho hình lập phương (ABCD.A'B'C'D'). Góc giữa hai vectơ (overrightarrow {AC} ,overrightarrow {DA'} ) bằng: A. ({30^ circ }). B. ({45^ circ }). C. ({120^ circ }). D. ({60^ circ }).
  • 🧜 Giải bài 6 trang 60 sách bài tập toán 12 - Cánh diều Trong không gian, cho hai vectơ (overrightarrow a ,overrightarrow b ) tạo với nhau một góc ({60^ circ }) và (left| {overrightarrow a } right| = 3cm,left| {overrightarrow b } right| = 4cm). Khi đó (overrightarrow a .overrightarrow b ) bằng: A. 12. B. 6. C. (6sqrt 3 ). D. ‒6.
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