Tích có hướng và ứng dụng !!

A.một véc tơ

B.một số thực

C.một véc tơ khác \[\vec 0\]

D.một số thực khác 0

A.\[\vec u = \left( {\left| {\begin{array}{*{20}{c}}{\begin{array}{*{20}{l}}{{y_2}}\\{{y_1}}\end{array}}&{\begin{array}{*{20}{l}}{{z_2}}\\{{z_1}}\end{array}}\end{array}} \right|;\left| {\begin{array}{*{20}{c}}{\begin{array}{*{20}{l}}{{z_2}}\\{{z_1}}\end{array}}&{\begin{array}{*{20}{l}}{{x_2}}\\{{x_1}}\end{array}}\end{array}} \right|;\left| {\begin{array}{*{20}{c}}{\begin{array}{*{20}{l}}{{x_2}}\\{{x_1}}\end{array}}&{\begin{array}{*{20}{l}}{{y_2}}\\{{y_1}}\end{array}}\end{array}} \right|} \right)\]

B. \[\vec u = \left( {\left| {\begin{array}{*{20}{c}}{\begin{array}{*{20}{l}}{{x_1}}\\{{x_2}}\end{array}}&{\begin{array}{*{20}{l}}{{y_1}}\\{{y_2}}\end{array}}\end{array}} \right|;\left| {\begin{array}{*{20}{c}}{\begin{array}{*{20}{l}}{{y_1}}\\{{y_2}}\end{array}}&{\begin{array}{*{20}{l}}{{z_1}}\\{{z_2}}\end{array}}\end{array}} \right|;\left| {\begin{array}{*{20}{c}}{\begin{array}{*{20}{l}}{{z_1}}\\{{z_2}}\end{array}}&{\begin{array}{*{20}{l}}{{x_1}}\\{{x_2}}\end{array}}\end{array}} \right|} \right)\]

C. \[\vec u = \left( {\left| {\begin{array}{*{20}{c}}{\begin{array}{*{20}{l}}{{y_1}}\\{{y_2}}\end{array}}&{\begin{array}{*{20}{l}}{{z_1}}\\{{z_2}}\end{array}}\end{array}} \right|;\left| {\begin{array}{*{20}{c}}{\begin{array}{*{20}{l}}{{z_1}}\\{{z_2}}\end{array}}&{\begin{array}{*{20}{l}}{{x_1}}\\{{x_2}}\end{array}}\end{array}} \right|;\left| {\begin{array}{*{20}{c}}{\begin{array}{*{20}{l}}{{x_1}}\\{{x_2}}\end{array}}&{\begin{array}{*{20}{l}}{{y_1}}\\{{y_2}}\end{array}}\end{array}} \right|} \right)\]

D. \[\vec u = \left( {\left| {\begin{array}{*{20}{c}}{\begin{array}{*{20}{l}}{{z_1}}\\{{z_2}}\end{array}}&{\begin{array}{*{20}{l}}{{x_1}}\\{{x_2}}\end{array}}\end{array}} \right|;\left| {\begin{array}{*{20}{c}}{\begin{array}{*{20}{l}}{{x_1}}\\{{x_2}}\end{array}}&{\begin{array}{*{20}{l}}{{y_1}}\\{{y_2}}\end{array}}\end{array}} \right|;\left| {\begin{array}{*{20}{c}}{\begin{array}{*{20}{l}}{{y_1}}\\{{y_2}}\end{array}}&{\begin{array}{*{20}{l}}{{z_1}}\\{{z_2}}\end{array}}\end{array}} \right|} \right)\]

A.\[\vec 0\]

B. \[\left( { - 2; - 1; - 1} \right)\]

C. \[\left( {2;1;1} \right)\]

D. \[\left( { - 1; - 2; - 1} \right)\]

A.\[\left[ {\overrightarrow {{u_1}} ,\overrightarrow {{u_2}} } \right] = \left[ {\overrightarrow {{u_2}} ,\overrightarrow {{u_1}} } \right]\]

B. \[\left[ {\overrightarrow {{u_1}} ,\overrightarrow {{u_2}} } \right] = - \left[ {\overrightarrow {{u_2}} ,\overrightarrow {{u_1}} } \right]\]

C. \[\left[ {\overrightarrow {{u_1}} ,\overrightarrow {{u_2}} } \right] - \left[ {\overrightarrow {{u_2}} ,\overrightarrow {{u_1}} } \right] = \vec 0\]

D. \[\left[ {\overrightarrow {{u_1}} ,\overrightarrow {{u_2}} } \right] + \left[ {\overrightarrow {{u_2}} ,\overrightarrow {{u_1}} } \right] = 0\]

A.\[\overrightarrow {{u_1}} .\overrightarrow {{u_2}} = 0\]

B. \[\overrightarrow {{u_1}} .\overrightarrow {{u_2}} = \vec 0\]

C. \[\left[ {\overrightarrow {{u_1}} ,\overrightarrow {{u_2}} } \right] = \vec 0\]

D. \[\left[ {\overrightarrow {{u_1}} ,\overrightarrow {{u_2}} } \right] = 0\]

A.\[\overrightarrow {{u_1}} = k\overrightarrow {{u_2}} \]

B. \[\overrightarrow {{u_2}} = k\overrightarrow {{u_1}} \]

C. \[\left[ {\overrightarrow {{u_1}} ,\overrightarrow {{u_2}} } \right] = \vec 0\]

D. \[\overrightarrow {{u_1}} .\overrightarrow {{u_2}} = \vec 0\]

A.b=2c

B.c=2b

C.b=−2c

D.b=c

A.\[\left[ {\overrightarrow {{u_1}} ;\overrightarrow {{u_2}} } \right].\overrightarrow {{u_1}} = 0\]

B. \[\left[ {\overrightarrow {{u_1}} ;\overrightarrow {{u_2}} } \right].\overrightarrow {{u_2}} = \vec 0\]

C. \[\left[ {\overrightarrow {{u_1}} ;\overrightarrow {{u_2}} } \right].\overrightarrow {{u_2}} = 0\]

D. \[\left[ {\overrightarrow {{u_1}} ;\overrightarrow {{u_2}} } \right] \bot \overrightarrow {{u_1}} \]

A.đồng phẳng

B.đôi một vuông góc     

C.cùng phương

D.cùng hướng

A.\[\left| {\left[ {\overrightarrow {{u_1}} ;\overrightarrow {{u_2}} } \right]} \right| = \left| {\overrightarrow {{u_1}} } \right|.\left| {\overrightarrow {{u_2}} } \right|\sin \left( {\overrightarrow {{u_1}} ,\overrightarrow {{u_2}} } \right)\]

B. \[\left| {\left[ {\overrightarrow {{u_1}} ;\overrightarrow {{u_2}} } \right]} \right| = \left| {\overrightarrow {{u_1}} } \right|.\left| {\overrightarrow {{u_2}} } \right|\cos \left( {\overrightarrow {{u_1}} ,\overrightarrow {{u_2}} } \right)\]

C. \[\left[ {\overrightarrow {{u_1}} ;\overrightarrow {{u_2}} } \right] = \left| {\overrightarrow {{u_1}} } \right|.\left| {\overrightarrow {{u_2}} } \right|\sin \left( {\overrightarrow {{u_1}} ,\overrightarrow {{u_2}} } \right)\]

D. \[\left| {\left[ {\overrightarrow {{u_1}} ;\overrightarrow {{u_2}} } \right]} \right| = \left| {\overrightarrow {{u_1}} .\overrightarrow {{u_2}} } \right|\sin \left( {\overrightarrow {{u_1}} ,\overrightarrow {{u_2}} } \right)\]

A.\[\sin \left( {\overrightarrow {{u_1}} ,\overrightarrow {{u_2}} } \right) = \frac{{\left| {\left[ {\overrightarrow {{u_1}} ;\overrightarrow {{u_2}} } \right]} \right|}}{{\left| {\overrightarrow {{u_1}} } \right|.\left| {\overrightarrow {{u_2}} } \right|}}\]

B. \[\sin \left( {\overrightarrow {{u_1}} ,\overrightarrow {{u_2}} } \right) = \frac{{\left[ {\overrightarrow {{u_1}} ;\overrightarrow {{u_2}} } \right]}}{{\left| {\overrightarrow {{u_1}} } \right|.\left| {\overrightarrow {{u_2}} } \right|}}\]

C. \[\sin \left( {\overrightarrow {{u_1}} ,\overrightarrow {{u_2}} } \right) = \frac{{\overrightarrow {{u_1}} .\overrightarrow {{u_2}} }}{{\left| {\overrightarrow {{u_1}} } \right|.\left| {\overrightarrow {{u_2}} } \right|}}\]

D. \[\sin \left( {\overrightarrow {{u_1}} ,\overrightarrow {{u_2}} } \right) = \frac{{\left| {\overrightarrow {{u_1}} .\overrightarrow {{u_2}} } \right|}}{{\left| {\overrightarrow {{u_1}} } \right|.\left| {\overrightarrow {{u_2}} } \right|}}\]

A.1     

B.\[\sqrt {\frac{{19}}{{26}}} \]

C. \[\sqrt {\frac{1}{2}} \]

D. \[\sqrt {\frac{{17}}{{26}}} \]

A.\[{S_{ABC}} = \left| {\left[ {\overrightarrow {AB} ,\overrightarrow {AC} } \right]} \right|\]

B. \[{S_{ABC}} = \frac{1}{2}\left| {\left[ {\overrightarrow {AB} ,\overrightarrow {AC} } \right]} \right|\]

C. \[{S_{ABC}} = \frac{1}{4}\left| {\left[ {\overrightarrow {AB} ,\overrightarrow {AC} } \right]} \right|\]

D. \[{S_{ABC}} = \frac{1}{6}\left| {\left[ {\overrightarrow {AB} ,\overrightarrow {AC} } \right]} \right|\]

A.\(\sqrt 5 \)

B.4

C.\(2\sqrt 5 \)

D.2

A.\[{S_{ABCD}} = \left| {\left[ {\overrightarrow {AB} ,\overrightarrow {AD} } \right]} \right|\]

B. \[{S_{ABCD}} = \left| {\left[ {\overrightarrow {AB} ,\overrightarrow {AC} } \right]} \right|\]

C. \[{S_{ABCD}} = \left| {\left[ {\overrightarrow {BC} ,\overrightarrow {BD} } \right]} \right|\]

D. \[{S_{ABCD}} = \left| {\left[ {\overrightarrow {AC} ,\overrightarrow {BD} } \right]} \right|\]

A.\(\sqrt 5 \)

B. \(2\sqrt 5 \)

C. \(2\sqrt 6 \)

D. \(2\sqrt 2 \)

A.\[{V_{ABCD}} = \frac{1}{3}\left| {\left[ {\overrightarrow {AB} ,\overrightarrow {AC} } \right].\overrightarrow {AD} } \right|\]

B. \[{V_{ABCD}} = \frac{1}{6}\left| {\left[ {\overrightarrow {AB} ,\overrightarrow {AD} } \right].\overrightarrow {AD} } \right|\]

C. \[{V_{ABCD}} = \frac{1}{6}\left| {\left[ {\overrightarrow {AB} ,\overrightarrow {AC} } \right].\overrightarrow {AD} } \right|\]

D. \[{V_{ABCD}} = \frac{1}{6}\left| {\left[ {\overrightarrow {AB} ,\overrightarrow {AD} } \right].\overrightarrow {AB} } \right|\]

A.1

B.6

C.3

D.2

A.\[{V_{ABCD.A'B'C'D'}} = \left[ {\overrightarrow {AB} ,\overrightarrow {AD} } \right].\overrightarrow {AA'} \]

B. \[{V_{ABCD.A'B'C'D'}} = \frac{1}{6}\left| {\left[ {\overrightarrow {AB} ,\overrightarrow {AD} } \right].\overrightarrow {AA'} } \right|\]

C. \[{V_{ABCD.A'B'C'D'}} = \left| {\left[ {\overrightarrow {AB} ,\overrightarrow {AD} } \right].\overrightarrow {AA'} } \right|\]

D. \[{V_{ABCD.A'B'C'D'}} = \frac{1}{3}\left| {\left[ {\overrightarrow {AB} ,\overrightarrow {AD} } \right].\overrightarrow {AA'} } \right|\]Trả lời:

A.1

B.Vô số

C.0

D.2

A.\[\vec w = \left( {1;7;1} \right).\]

B. \[\vec w = \left( { - 1;7; - 1} \right).\]

C. \[\vec w = \left( {0;7;1} \right).\]

D. \[\vec w = \left( {0; - 1;0} \right).\]

D.\(\frac{1}{6}\)

C. $\overrightarrow{u}=\left(\left|\begin{array}{cc}\begin{array}{l}{y}_1\\ y_2\end{array}& \begin{array}{l}{z}_1\\ z_2\end{array}\end{array}\right|;\left|\begin{array}{cc}\begin{array}{l}{z}_1\\ z_2\end{array}& \begin{array}{l}{x}_1\\ x_2\end{array}\end{array}\right|;\left|\begin{array}{cc}\begin{array}{l}{x}_1\\ x_2\end{array}& \begin{array}{l}{y}_1\\ y_2\end{array}\end{array}\right|\right)$

C. $\left[\overrightarrow{u_1},\overrightarrow{u_2}\right]-\left[\overrightarrow{u_2},\overrightarrow{u_1}\right]=\overrightarrow{0}$

C. $\left[\overrightarrow{u_1},\overrightarrow{u_2}\right]=\overrightarrow{0}$

C. $\left[\overrightarrow{u_1};\overrightarrow{u_2}\right].\overrightarrow{u_2}=0$

C. $\left[\overrightarrow{u_1};\overrightarrow{u_2}\right]=\left|\overrightarrow{u_1}\right|.\left|\overrightarrow{u_2}\right|\sin \left(\overrightarrow{u_1},\overrightarrow{u_2}\right)$

C. $S_{ABC}=\frac{1}{4}\left|\left[\overrightarrow{AB},\overrightarrow{AC}\right]\right|$

C. $S_{ABCD}=2\left|\left[\overrightarrow{AB},\overrightarrow{AD}\right]\right|$

A.$V_{ABCD.A^{\text{'}}{B}^{\text{'}}{C}^{\text{'}}{D}^{\text{'}}}=\left[\overrightarrow{AB},\overrightarrow{AD}\right].\overrightarrow{A{A}^{\text{'}}}$

C. $V_{ABCD.A^{\text{'}}{B}^{\text{'}}{C}^{\text{'}}{D}^{\text{'}}}=\left|\left[\overrightarrow{AB},\overrightarrow{AD}\right].\overrightarrow{A{A}^{\text{'}}}\right|$

C.(2;1;1)

C.b=−2c

C.cùng phương

C. $\sin \left(\overrightarrow{u_1},\overrightarrow{u_2}\right)=\frac{\overrightarrow{u_1}.\overrightarrow{u_2}}{\left|\overrightarrow{u_1}\right|.\left|\overrightarrow{u_2}\right|}$

C. $\sqrt{\frac{1}{2}}$

C. $2\sqrt{5}$

B. $S_{ABCD}=\left|\left[\overrightarrow{AB},\overrightarrow{AC}\right]\right|$

C. $S_{ABCD}=\left|\left[\overrightarrow{BC},\overrightarrow{BD}\right]\right|$

C. $2\sqrt{6}$

C.3

C. $\overrightarrow{w}=\left(0;7;1\right).$

C.0