Trong không gian với hệ tọa độ \(Oxyz\), cho hai véc tơ \(\overrightarrow a = 2\overrightarrow i - \overrightarrow j + 3\overrightarrow k \) và \(\overrightarrow b = 5\overrightarrow i + 3\overrightarrow j - 2\overrightarrow k \).              

Khi đó \(\overrightarrow {2a} .\left( { - 3\overrightarrow b } \right)\) bằng:

Gọi \[M\left( {a\,;0\,;0} \right),N\left( {0\,;b\,;0} \right),P\left( {0\,;0\,;\,c} \right)\].

\[\begin{array}{l}\overrightarrow {DM}  = \left( {a - 4;1; - 3} \right)\\\overrightarrow {DN}  = \left( { - 4;b + 1; - 3} \right)\\\overrightarrow {DP}  = \left( { - 4;1;c - 3} \right)\end{array}\]

Ta có \[DM,DN,DP\] đôi một vuông góc với nhau nên

\[\left\{ \begin{array}{l}\overrightarrow {DM} .\overrightarrow {DN}  = 0\\\overrightarrow {DM} .\overrightarrow {DP}  = 0\\\overrightarrow {DN} .\overrightarrow {DP}  = 0\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} - 4\left( {a - 4} \right) + b + 1 + 9 = 0\\ - 4\left( {a - 4} \right) + 1 - 3\left( {c - 3} \right) = 0\\16 + b + 1 - 3\left( {c - 3} \right) = 0\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} - 4a + b =  - 26\\ - 4a - 3c =  - 26\\b - 3c =  - 26\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}a = \frac{{13}}{4}\\b =  - 13\\c = \frac{{13}}{3}\end{array} \right.\].

a) Sai.

b) Sai.

c) Đúng vì: \[{V_{DMNP}} = \frac{1}{6}DM.DN.DP = \frac{1}{6}.\frac{{13}}{4}.13.\frac{{13}}{3} = \frac{{2197}}{{72}} > 29\].

d) Gọi \[\overrightarrow x  = \left( {m\,;n\,;p} \right)\]

\[\overrightarrow {DM}  = \left( { - \frac{3}{4};\,1\,; - 3} \right);\,\,\overrightarrow {DN}  = \left( { - 4; - 12; - 3} \right);\,\,\,\overrightarrow {DP}  = \left( { - 4;1\,;\,\frac{4}{3}} \right)\]

\[\left\{ \begin{array}{l}\overrightarrow x .\overrightarrow {DM}  = 1\\\overrightarrow x .\overrightarrow {DN}  = 2\\\overrightarrow x .\overrightarrow {DP}  =  - 3\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} - \frac{3}{4}m + n - 3p = 1\\ - 4m - 12n - 3p = 2\\ - 4m + n + \frac{4}{3}p =  - 3\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}m = \frac{{88}}{{169}}\\n =  - \frac{{35}}{{169}}\\p =  - \frac{{90}}{{169}}\end{array} \right.\]

\[m\, + n\, + p = \frac{{ - 37}}{{169}}\].

Suy ra d) sai.

Trả lời: \[\left| {\overrightarrow a  - 2\overrightarrow b } \right| \approx 15,7\].

Ta có: \[\overrightarrow a  = 2\overrightarrow i  - 3\overrightarrow j  + 6\overrightarrow k \] \( \Rightarrow \overrightarrow a  = \left( {2; - 3;6} \right)\)

\[\overrightarrow b  = 6\overrightarrow j  + \overrightarrow k \]\( \Rightarrow \overrightarrow b  = \left( {0;6;1} \right)\)

Khi đó: \[\overrightarrow a  - 2\overrightarrow b  = \left( {2; - 15;4} \right) \Rightarrow \left| {\overrightarrow a  - 2\overrightarrow b } \right| = 7\sqrt 5  \approx 15,7.\]