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Giải bài tập 2.12 trang 59 SGK Toán 12 tập 1 - Kết nối tri thức

Cho tứ diện ABCD. Chứng minh rằng: a) (overrightarrow {AB} .overrightarrow {CD} = overrightarrow {AC} .overrightarrow {CD} + overrightarrow {BC} .overrightarrow {DC} ); b) (overrightarrow {AB} .overrightarrow {CD} + overrightarrow {AC} .overrightarrow {DB} + overrightarrow {AD} .overrightarrow {BC} = 0).

🦩Tổng hợp đề thi học kì 2 lớp 12 tất cả các môn - Kết nối tri thức

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

Cho tứ diện ABCD. Chứng minh rằng:
a) \(\overrightarrow {AB} .\overrightarrow {CD} = \overrightarrow {AC} .\overrightarrow {CD} + \overrightarrow {BC} .\overrightarrow {DC} \);
🏅b) \(\overrightarrow {AB} .\overrightarrow {CD} + \overrightarrow {AC} .\overrightarrow {DB} + \overrightarrow {AD} .\overrightarrow {BC} = 0\).

Phương pháp giải - Xem chi tiết

Sử dụng kiến thức về quy tắc ba điểm để chứng minh: Nếu A, B, C là ba điểm bất kì thì \(\overrightarrow {AB}  + \overrightarrow {BC}  = \overrightarrow {AC} \).

Lời giải chi tiết

a) Ta có: \(\overrightarrow {AC} .\overrightarrow {CD}  + \overrightarrow {BC} .\overrightarrow {DC}  = \overrightarrow {AC} .\overrightarrow {CD}  - \overrightarrow {BC} .\overrightarrow {CD}  = \overrightarrow {CD} \left( {\overrightarrow {AC}  + \overrightarrow {CB} } \right) = \overrightarrow {CD} .\overrightarrow {AB} \) (đpcm) b) \(\overrightarrow {AB} .\overrightarrow {CD}  + \overrightarrow {AC} .\overrightarrow {DB}  + \overrightarrow {AD} .\overrightarrow {BC}  = \overrightarrow {AB} .\overrightarrow {CD}  + \left( {\overrightarrow {AB}  + \overrightarrow {BC} } \right).\overrightarrow {DB}  + \left( {\overrightarrow {AB}  + \overrightarrow {BD} } \right).\overrightarrow {BC} \) \( = \overrightarrow {AB} .\overrightarrow {CD}  + \overrightarrow {AB} .\overrightarrow {DB}  + \overrightarrow {BC} .\overrightarrow {DB}  + \overrightarrow {AB} .\overrightarrow {BC}  + \overrightarrow {BD} .\overrightarrow {BC} \) \( = \overrightarrow {AB} .\left( {\overrightarrow {CD}  + \overrightarrow {DB}  + \overrightarrow {BC} } \right) + \left( {\overrightarrow {BC} .\overrightarrow {DB}  + \overrightarrow {BD} .\overrightarrow {BC} } \right) = \overrightarrow {AB} .\left( {\overrightarrow {CB}  + \overrightarrow {BC} } \right) + \overrightarrow {BC} \left( {\overrightarrow {DB}  + \overrightarrow {BD} } \right) = 0\)

  • 🍃 Giải bài tập 2.11 trang 59 SGK Toán 12 tập 1 - Kết nối tri thức Trong không gian, cho hai vectơ (overrightarrow a ) và (overrightarrow b ) có cùng độ dài bằng 1. Biết rằng góc giữa hai vectơ đó là ({45^0}), hãy tính: a) (overrightarrow a .overrightarrow b ); b) (left( {overrightarrow a + 3overrightarrow b } right).left( {overrightarrow a - 2overrightarrow b } right)) c) ({left( {overrightarrow a + overrightarrow b } right)^2}).
  • 🐬 Giải bài tập 2.10 trang 59 SGK Toán 12 tập 1 - Kết nối tri thức Cho hình lăng trụ tứ giác đều ABCD.A’B’C’D’ có độ dài mỗi cạnh đáy bằng 1 và độ dài mỗi cạnh bên bằng 2. Hãy tính góc giữa các cặp vectơ sau đây và tính tích vô hướng của mỗi cặp vectơ đó: a) (overrightarrow {AA'} ) và (overrightarrow {C'C;} ) b) (overrightarrow {AA'} ) và (overrightarrow {BC;} ) c) (overrightarrow {AC} ) và (overrightarrow {B'A'} ).
  • ꦓ Giải bài tập 2.9 trang 59 SGK Toán 12 tập 1 - Kết nối tri thức Ba sợi dây không giãn với khối lượng không đáng kể được buộc chung một đầu và được kéo căng về ba hướng khác nhau (H.2.31). Nếu các lực kéo làm cho ba sợi dây ở trạng thái đứng yên thì khi đó ba sợi dây nằm trên cùng một mặt phẳng. Hãy giải thích vì sao.
  • 🦩 Giải bài tập 2.8 trang 58 SGK Toán 12 tập 1 - Kết nối tri thức Trong Luyện tập 8, ta đã biết trọng tâm của tứ diện ABCD là một điểm I thỏa mãn (overrightarrow {AI} = 3overrightarrow {IG} ), ở đó G là trọng tâm của tam giác BCD. Áp dụng tính chất trên để tính khoảng cách từ trọng tâm của một khối rubik (đồng chất) hình tứ diện đều đến một mặt của nó, biết rằng chiều cao của khối rubik là 8cm (H.2.30).
  • 🎐 Giải bài tập 2.7 trang 58 SGK Toán 12 tập 1 - Kết nối tri thức Cho hình chóp S.ABC. Trên cạnh SA, lấy điểm M sao cho \(SM = 2AM\). Trên cạnh BC, lấy điểm N sao cho \(CN = 2BN\). Chứng minh rằng \(\overrightarrow {MN} = \frac{1}{3}\left( {\overrightarrow {SA} + \overrightarrow {BC} } \right) + \overrightarrow {AB} \).
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