Bài 9 trang 117 Sách bài tập Hình học lớp 12 Nâng cao

\(\eqalign{  & a)\overrightarrow u  = (1;2; - 3),\overrightarrow v  = ( - 4;1;2);  \cr  & b)\overrightarrow u  = 3\overrightarrow i  + 2\overrightarrow j  - \overrightarrow k ,\overrightarrow v  =  - \overrightarrow i  - 3\overrightarrow j  + \overrightarrow k ;  \cr  & c)\overrightarrow u  = (0;1; - 2),\overrightarrow v  = (3;0; - 4)  \cr  & d)\overrightarrow u  = 4\overrightarrow i  + \overrightarrow k ,\overrightarrow v  = 2\overrightarrow i  - \overrightarrow {j;}  \cr} \)

Giải

\(\eqalign{  & a)\left[ {\overrightarrow u ,\overrightarrow v } \right] = \left( {\left| \matrix{  2 \hfill \cr  1 \hfill \cr}  \right.\left. \matrix{   - 3 \hfill \cr  2 \hfill \cr}  \right|;\left| \matrix{   - 3 \hfill \cr  2 \hfill \cr}  \right.\left. \matrix{  1 \hfill \cr   - 4 \hfill \cr}  \right|;\left| \matrix{  1 \hfill \cr   - 4 \hfill \cr}  \right.\left. \matrix{  2 \hfill \cr  1 \hfill \cr}  \right|} \right) \cr&= (7;10;9).  \cr  & b)\overrightarrow u  = (3;2; - 1),\overrightarrow v  = ( - 1; - 3;1)  \cr  &  \Rightarrow \left[ {\overrightarrow u ,\overrightarrow v } \right] = \left( {\left| \matrix{  2 \hfill \cr   - 3 \hfill \cr}  \right.\left. \matrix{   - 1 \hfill \cr  1 \hfill \cr}  \right|;\left| \matrix{   - 1 \hfill \cr  1 \hfill \cr}  \right.\left. \matrix{  3 \hfill \cr   - 1 \hfill \cr}  \right|;\left| \matrix{  3 \hfill \cr   - 1 \hfill \cr}  \right.\left. \matrix{  2 \hfill \cr   - 3 \hfill \cr}  \right|} \right)\cr& = ( - 1; - 2; - 7).  \cr  & c)\left[ {\overrightarrow u ,\overrightarrow v } \right] = \left( {\left| \matrix{  1 \hfill \cr  0 \hfill \cr}  \right.\left. \matrix{   - 2 \hfill \cr   - 4 \hfill \cr}  \right|;\left| \matrix{   - 2 \hfill \cr   - 4 \hfill \cr}  \right.\left. \matrix{  0 \hfill \cr  3 \hfill \cr}  \right|;\left| \matrix{  0 \hfill \cr  3 \hfill \cr}  \right.\left. \matrix{  1 \hfill \cr  0 \hfill \cr}  \right|} \right) \cr&= ( - 4; - 6; - 3).  \cr  & d)\overrightarrow u  = (4;0;1),\overrightarrow v  = (2; - 1;0)  \cr  &  \Rightarrow \left[ {\overrightarrow u ,\overrightarrow v } \right] = \left( {\left| \matrix{  0 \hfill \cr   - 1 \hfill \cr}  \right.\left. \matrix{  1 \hfill \cr  0 \hfill \cr}  \right|;\left| \matrix{  1 \hfill \cr  0 \hfill \cr}  \right.\left. \matrix{  4 \hfill \cr  2 \hfill \cr}  \right|;\left| \matrix{  4 \hfill \cr  2 \hfill \cr}  \right.\left. \matrix{  0 \hfill \cr   - 1 \hfill \cr}  \right|} \right) \cr&= (1;2; - 4). \cr} \)

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