جواب تمرین صفحه 76 درس 5 ریاضی هشتم (بردار و مختصات)

\(\begin{array}{l}( - 1)\left[ {\begin{array}{*{20}{c}}3\\5\end{array}} \right] - \left[ {\begin{array}{*{20}{c}}3\\5\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{ - 3}\\{ - 5}\end{array}} \right] - \left[ {\begin{array}{*{20}{c}}3\\5\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{ - 6}\\{ - 10}\end{array}} \right]\\\\( - 4)\left[ {\begin{array}{*{20}{c}}{ - 5}\\7\end{array}} \right] - \left[ {\begin{array}{*{20}{c}}2\\{ - 1}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{20}\\{ - 28}\end{array}} \right] - \left[ {\begin{array}{*{20}{c}}2\\{ - 1}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{18}\\{ - 27}\end{array}} \right]\\\\( - \frac{1}{2})\left[ {\begin{array}{*{20}{c}}{12}\\{ - 8}\end{array}} \right] + 6\left[ {\begin{array}{*{20}{c}}7\\3\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{ - 6}\\4\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}{42}\\{18}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{36}\\{22}\end{array}} \right]\end{array}\)

\(\begin{array}{l}4x = \left[ {\begin{array}{*{20}{c}}{12}\\{ - 8}\end{array}} \right] \Rightarrow x = \frac{1}{4} \times \left[ {\begin{array}{*{20}{c}}{12}\\{ - 8}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}3\\{ - 2}\end{array}} \right]\\\\ - 3x = \left[ {\begin{array}{*{20}{c}}{15}\\{ - 9}\end{array}} \right] \Rightarrow x = \frac{{ - 1}}{3} \times \left[ {\begin{array}{*{20}{c}}{15}\\{ - 9}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{ - 5}\\3\end{array}} \right]\\\\\left[ {\begin{array}{*{20}{c}}2\\5\end{array}} \right] + x = \left[ {\begin{array}{*{20}{c}}0\\0\end{array}} \right] \Rightarrow x = \left[ {\begin{array}{*{20}{c}}0\\0\end{array}} \right] - \left[ {\begin{array}{*{20}{c}}2\\5\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{ - 2}\\{ - 5}\end{array}} \right]\\\\\left[ {\begin{array}{*{20}{c}}2\\5\end{array}} \right] + x = \left[ {\begin{array}{*{20}{c}}{ - 4}\\6\end{array}} \right] \Rightarrow x = \left[ {\begin{array}{*{20}{c}}{ - 4}\\6\end{array}} \right] - \left[ {\begin{array}{*{20}{c}}2\\5\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{ - 6}\\1\end{array}} \right]\end{array}\)

بردار W را به دو بردار a و b تجزیه کردیم:

\(\begin{array}{l}\left\{ \begin{array}{l}\vec a = \left[ {\begin{array}{*{20}{c}}2\\1\end{array}} \right]\,\,\,,\,\,\,\vec b = \left[ {\begin{array}{*{20}{c}}{ - 2}\\3\end{array}} \right]\\\\\vec c = \vec a - 2\vec b\end{array} \right.\\\\ \Rightarrow \vec c = \left[ {\begin{array}{*{20}{c}}2\\1\end{array}} \right] - 2\left[ {\begin{array}{*{20}{c}}{ - 2}\\3\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}6\\{ - 5}\end{array}} \right]\end{array}\)

\(\begin{array}{l}\left\{ \begin{array}{l}\vec a = \left[ {\begin{array}{*{20}{c}}1\\0\end{array}} \right]\,\,\,,\,\,\,\vec b = \left[ {\begin{array}{*{20}{c}}0\\1\end{array}} \right]\\\\\vec c = 2\vec a + 3\vec b\end{array} \right.\\\\ \Rightarrow \vec c = 2\left[ {\begin{array}{*{20}{c}}1\\0\end{array}} \right] + 3\left[ {\begin{array}{*{20}{c}}0\\1\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}2\\3\end{array}} \right]\end{array}\)

\(\begin{array}{l}\left\{ \begin{array}{l}\vec a = \left[ {\begin{array}{*{20}{c}}1\\0\end{array}} \right]\,\,\,,\,\,\,\vec b = \left[ {\begin{array}{*{20}{c}}0\\1\end{array}} \right]\\\\\vec c = - 3\vec a + 4\vec b\end{array} \right.\\\\ \Rightarrow \vec c = - 3\left[ {\begin{array}{*{20}{c}}1\\0\end{array}} \right] + 4\left[ {\begin{array}{*{20}{c}}0\\1\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{ - 3}\\4\end{array}} \right]\end{array}\)